1 Introduction↩︎

An oriented rational homology sphere \(M\) is an \(L\)-space if \(\widehat{\mathit{HF}}(M)\) has dimension equal to the cardinality \(|H_1(M; {\mathbb{Z}})|\) of its first homology group. Since this is the smallest possible rank, \(L\)-spaces have minimal Heegaard Floer homology. Standard examples include lens spaces and, more generally, elliptic \(3\)-manifolds [1]. There are also many hyperbolic examples: for instance, every \((r_1,r_2,r_3)\)-surgery on the Borromean rings, with \(r_i\geq 1\), is an \(L\)-space. Although defined purely in terms of Heegaard Floer homology, \(L\)-spaces have been a useful in studying several different topics in low-dimensional topology. For example, they can be used to study unknotting numbers ([2], [3]), the cosmetic crossing conjecture ([4]), and to prove vanishing results for Seiberg-Witten invariants ([5], [6]). They also do not support coorientable taut foliations ([7]–[9]), and are conjecturally characterized among irreducible \(3\)-manifolds by this property; see [10]. In a different direction, Boyer, Gordon and Watson conjectured in [11] that an irreducible \(3\)-manifold is an \(L\)-spaces if and only if its fundamental group is not left-orderable. The conjectured equivalence of these three notions is now known as the \(L\)-space conjecture and it has been proven for graph manifolds [12]–[14], but remains open in general.

A feature of Heegaard Floer theory is that information about the Heegaard Floer homology of a \(3\)-manifold \(M\) can often be extracted from a presentation of \(M\) as surgery on a link. This is especially effective when determining whether \(M\) is an \(L\)-space. In [15], the definition of an \(L\)-space link was introduced.

This definition can be naturally extended to that of generalised \((\pm,\dots,\pm)\,L\)-space links, obtained by considering the corresponding types of generalised large surgeries. According to this definition, an \(L\)-space link is a generalised \((+,\dots, +)\,L\)-space link.

There is a significant difference between studying \(L\)-space surgeries on knots and on links with two or more components. For example, if a knot \(K\) in \(S^3\) has a non-meridional \(L\)-space surgery, then, up to mirroring, \(K\) is an \(L\)-space knot. More precisely, the \(r\)-surgery on such a knot \(K\) is an \(L\)-space if and only if \(r\in [2g(K)-1, \infty]\) [16], where \(g(K)\) denotes the genus of \(K\) and \(r\in \overline{ {\mathbb{Q}}}= {\mathbb{Q}}\cup \{\infty\}\). In addition, \(L\)-space knots satisfy strong topological constraints: they are fibered with right-veering monodromy, they are strongly quasipositive, and their \(3\)-dimensional and \(4\)-dimensional genus coincide and agree with their \(\tau\) invariant [17]–[20]. For links, many of these properties do not hold. In particular, \(L\)-space links need not be fibered [21], or strongly quasipositive [22], and there exist links with positive \(L\)-space surgeries that are not \(L\)-space links. Examples of such links will appear also in this paper.

In general, our understanding of the possible shapes and structural properties of the set of \(L\)-space surgeries on a link, considered as a subset of \(\overline{ {\mathbb{Q}}}^n\), is very limited. In [23], the authors provide sufficient conditions for this set to be bounded from below for two-component \(L\)-space links, and they completely characterise when this happens for two-component algebraic links. In [24], [25], among other results, the possible regions of integer \(L\)-space surgeries on two-component \(L\)-space links are studied, and, in some cases, classified. The first non-trivial examples of complete classifications of rational \(L\)-space surgeries are given in [26] where, using a gluing formula for \(L\)-spaces together with the classification result of [14], the author classifies the set \(L\)-space surgeries on satellites by torus links.

For hyperbolic links, very few examples of classifications are known, and in those cases the set of \(L\)-space (rational) surgeries is fairly simple, being either empty of union of finitely many rectangles in $ {}^n$; for instance, we have one rectangle for the Whitehead link, and two rectangles for the Borromean rings¹. The goal of this paper is to describe the sets of \(L\)-space surgeries on all hyperbolic links with two components belonging to the family of two-bridge links, and, by doing so, to provide more interesting examples of sets of \(L\)-space surgeries.

We refer to Section 2 for definitions and some properties of two-bridge links. For now, recall that two-bridge links can have either one or two components, and in this paper we will always consider two-component links, since for two-bridge knots the situation is well-understood². Two-bridge links have unknotted components, and hence any surgery where at least one surgery coefficient is \(\infty\) is either an \(L\)-space, indeed a lens space, or \(S^2\times S^1\). For this reason, we will focus only on non-trivial surgeries, i.e. those where no surgery coefficient is \(\infty\).

Our first result is a classification of two-bridge links with non-trivial \(L\)-space surgeries. Indeed, we prove a stronger statement:

The links \(L_{n,k}\) and \(L'_{n,k}\) are depicted in Figure 1, where we use boxes to indicate the number of crossings between two strands, with the convention that right-handed crossings are positive and left-handed crossings are negative. We observe that the families \(L_{n,k}\) and \(L'_{n,k}\), are not disjoint, and their intersection is given by the \(T(2,2k)\) torus links.

The bulk of the proof of Theorem 2 is presented in Section 3 and exploits diagrammatic properties of two-bridge links. More precisely, we introduce the notion of a persistently foliar link³, and we prove a general theorem which provides sufficient conditions for an \(n\)-component link to be persistently foliar; we refer to Section 3, and in particular to Theorem 30, for details. As a direct consequence of the definition, all non-trivial surgeries on a persistently foliar link support coorientable taut foliations, and since \(L\)-spaces do not, Theorem 30 allows us to identify the links in the families \(L_{n,k}\) and \(L'_{n,k}\) as the only two-bridge links that can have \(L\)-space surgeries. We show that this is indeed the case in Section 9.

The links \(L_{n,k}\) already appeared in the literature in [21], where Y. Liu conjectured that these are exactly the two-bridge \(L\)-space links. This was proved by Dawra in his thesis [30]. As a consequence of our results we obtain a different proof of Liu’s conjecture. In addition, with the next result, we classify all \(L\)-space surgeries on the links \(L_{n,k}\). Recall that a two-bridge link is either hyperbolic or a \(T(2,2k)\) torus link, with \(k\in {\mathbb{Z}}\). Dehn surgeries on torus links are Seifert fibered manifolds, or connected sums thereof, and since the \(L\)-spaces in this family are classified [31], we focus on the hyperbolic two-bridge links.

We are also able to complement Theorem 3 by showing that every non-\(L\)-space surgery on \(L_{n,k}\) supports a coorientable taut foliation.

We prove Theorems 3 and 4 in Section 9.1. When \(k=1\) or \(n=1\), the links \(L_{n,k}\) coincide with the fibered hyperbolic two-bridge links admitting \(L\)-space surgeries and were previously studied in [27], [32]. Interestingly, while in [27], [32] the first author used taut foliations to show that a certain explicitly described collection of surgeries on \(L_{n,1}\) constituted the complete set of \(L\)-space surgeries on \(L_{n,1}\), here the direction of the argument is partially reversed. Indeed, some of the taut foliations provided by Theorem 4 are obtained as a consequence of the classification result Theorem 3. This is made possible by Lyu’s recent study of persistently foliar \((1,1)\) non-\(L\)-space knots [33].

Focusing on the links \(L'_{n,k}\), we show in Section 9 that they are generalised \((+,-)\,L\)-space links and that, except for the torus links, they are not \(L\)-space links. Since two-bridge links are symmetric, i. e. they admit an isotopy exchanging their components, a two-bridge link is a generalised \((+,-)\,L\)-space link if and only if it is a generalised \((-,+)\,L\)-space link. We classify the \(L\)-space surgeries on the hyperbolic links in \(L'_{n,k}\) with the following theorem.

Although not immediately apparent, the statement of Theorem 5 is invariant under exchanging \(r\) and \(s\), reflecting the symmetry of two-bridge links. We show in Figure 3 the set of \(L\)-space surgeries for the link \(L'_{3,1}\), which exhibits the features of the general case.

The statement of Theorem 5 can be rephrased more concisely in terms of box-convex sets. We refer to Section 8.3 and specifically to Theorem 93 for this reformulation, and for more comments regarding the structure of the set of \(L\)-space surgeries on these links.

The proof of Theorem 5 is presented in Section 9.2 and differs substantially from the one of Theorem 3. In fact, the links \(L_{n,k}\) are \(L\)-space links, and hence we can take advantage of some simplifications in the rational link surgery formula, that we describe in more detail below. On the other hand, the hyperbolic links in the family \(L'_{n,k}\) are not \(L\)-space link; in this case we proceed with an explicit and detailed study of the Turaev torsions of the manifolds obtained by filling only one of the boundary components of the exterior of the link.

1.1 Rational link surgery formula and the simplified model↩︎

In the proof of Theorem 3, we obstruct certain rational surgeries on \(L_{n,k}\) from yielding \(L\)-spaces. This is achieved using a simplified model of the rational link surgery formula, following [21], [34], [35].

Given an oriented \(n\)-component link \(L\subset S^3\) and an integer framing \(\Lambda = (d_1,\dots,d_n)\), Manolescu and Ozsváth constructed in [34] a link surgery complex \(C_\Lambda(L)\) from the link Floer complex of \(L\) and proved that the homology of \(C_\Lambda(L)\) is the completed Heegaard Floer homology \(\boldsymbol{\mathbf{HF}}^{-}(S^3_{\Lambda}(L))\). One can similarly recover \(\widehat{\mathit{HF}}(S^3_{\Lambda}(L))\) from \(C_\Lambda(L)\). The construction is referred to as the (integral) link surgery formula.

Zemke reinterpreted the link surgery formula in the language of bordered modules in [35]. As a direct consequence, a version of the link surgery formula which allows for rational surgeries can be defined; see [35]. Since an explicit description of this rational link surgery formula has not appeared in the literature, we provide the details in Section 6.4 Specifically, we define a (rational) link surgery complex by extending the construction of \(C_\Lambda(L)\) to allow rational surgery coefficients, and verify that this complex is chain homotopy equivalent to \(\boldsymbol{\mathbf{CF}}^{-}(S^3_{\Lambda}(L))\); see Theorem 75.

In the case of integral framings, the link surgery formula admits certain simplifications under favorable conditions. By construction, \(C_\Lambda (L)\) is a hypercube of chain complexes. This means that its underlying module is a direct sum of subspaces \(C_\varepsilon\) for \(\varepsilon\in \{0,1\}^n\) and the differential is the sum of the maps \(D_{\varepsilon,\varepsilon'}\) for \(\varepsilon\leq \varepsilon'\). Y. Liu [21] defined a simplified model \(H_\Lambda(L)\) (referred to by Liu as the perturbed complex) by

• first replacing each \(C_\varepsilon\) by \(H_*(C_\varepsilon)\); and

• then retaining only the maps induced by the maps \(D_{\varepsilon,\varepsilon'}\) with \(|\varepsilon'-\varepsilon|=1\).

It was proved by Liu [21] that for surgeries with integral framings \(\Lambda\) on a two-component \(L\)-space link \(L\), the complex \(C_\Lambda(L)\) is homotopy equivalent to \(H_\Lambda(L)\). The proof relies on the fact that the link Floer homology of an \(L\)-space link is supported in even homological gradings. In a different setting, Zemke proved that, for integer surgeries on plumbed trees, \(H_\Lambda(L) \simeq C_\Lambda(L)\) in [36]. We extend Liu’s result on two-component \(L\)-space links to the setting of rational surgeries.

According to the simplified model, for a two-component \(L\)-space link \(L\), \(\boldsymbol{\mathbf{HF}}^{-}(S^3_\Lambda(L))\) is determined by the link Floer homology of \(L\), which in turn is determined by the multivariable Alexander polynomials of its sublinks via the so-called \(H\)-function [37]. Therefore to determine whether a surgery yields an \(L\)-space it suffices to study the \(H\)-function of \(L.\) This idea was used in [23], [25] to obstruct certain integer surgeries on \(L\) from being an \(L\)-space. Combining these arguments with the simplified model for rational link surgery, we are able to produce obstructions to \(L\)-space surgeries on \(L_{n,k}\) in Section 7.

1.2 Some consequences↩︎

We conclude the introduction by collecting some consequences of the main results of this paper.

1.2.1 Hyperbolic volumes↩︎

It follows from the main result of [38] that the set of volumes of hyperbolic two-bridge links is unbounded. This was already observed for fibered two-bridge links in [32], and it is not difficult to provide, in the same way, families of non-fibered two-bridge links whose volume goes to infinity. On the other hand, a direct consequence of Theorem 2 is that the volumes of hyperbolic \(L\)-spaces that are surgeries on two-bridge links are uniformly bounded. Let \(v_8\sim3.663862...\) denote the volume of the ideal regular hyperbolic octahedron.

1.2.2 Dehn surgery \((\)non-\()\)detection results↩︎

It is now well-known [17] that if \(K\subset S^3\) is a non-trivial knot and the manifold \(S^3_{1}(K)\), obtained by \(1\)-surgery on \(K\), is an \(L\)-space, then \(K\) is the right-handed trefoil knot. More recently, it was shown that if \(S^3_{2}(K)\) is an \(L\)-space, then \(K\) is either the trefoil or the cinquefoil [41].

For links with two components, such strong detection results are less common. On the other hand, the property that \((1,1)\)-surgery yields an \(L\)-space characterises the Whitehead link among two-component links with unknotted components and linking number zero [42], and among fibered hyperbolic two-bridge links [32]. Moreover, by using [25], [42] one obtains the following.

In contrast to the previous proposition, Theorem 5 implies that if one does not require \(L\) to be an \(L\)-space link, then there exist infinitely many hyperbolic links whose \((1,1)\)-surgery yields an \(L\)-space.

On the other hand, as consequence of Theorems 3 and 5, we can deduce the following detection results for the families \(L_{n,k}\) and \(L'_{n,k}\).

1.2.3 Satellite operations↩︎

We briefly recall the satellite operation. Let \(K\subset S^3\) be a knot, \(P \subset S^1 \times \mathbb{D}^2\) a knot in the solid torus, and let \(m\in \mathbb{Z}\) be an integer. We define a new knot \(P(K,m)\subset S^3\) by identifying \(S^1\times \mathbb{D}^2\) with a tubular neighbourhood of \(K\) in such a way that \(S^1 \times \{\theta\}\), where \(\theta\in \partial\mathbb{D}^2\), is identified to the \(m\)-framed longitude of \(K\). The knots \(K\) and \(P\) are called companion and pattern knot respectively, and \(P(K,m)\) is called a satellite knot.

If we view the solid torus containing \(P\) as the tubular neighbourhood of an unknot in \(S^3\), and denote by \(K_0\) its meridian, then we can naturally associate to the pattern knot the two-component link \(L_P=K_0 \cup P\subset S^3\). We call \(L_P\) the pattern link.

As a consequence of our main result, together with parts of the proofs presented in this paper, we obtain a classification of all \(L\)-space satellite knots whose pattern link is a two-bridge link.

Note that if the pattern \(L_P\) is a torus two-bridge link, i.e. a \(T(2,2l)\) torus link, then the satellite \(P(K,m)\) is a cable of \(K\), and \(L\)-space cable knots are completely classified by [16], [43]. For the proof of Theorem 12 we refer to Section 10.

Structure of the paper↩︎

The structure of the paper is as follows. The main body is divided into three parts: Sections 3–4, Sections 5–6–7, and Sections 8–9. We have tried to make these three parts as self-contained as possible, with minimal interdependence, so that the reader may approach each of them independently.

Sections 3–4 are devoted to the construction of taut foliations. More precisely, in Section 3 we prove Theorem 30, which, roughly speaking, states that every non-trivial surgery on an \(n\)-component link admitting a special diagram supports a coorientable taut foliation. Next, in Section 4, we apply this theorem, together with some variations of it, to the study of two-bridge links, proving the part of Theorem 2 concerning taut foliations. In the last part of this section, we also construct foliations on some surgeries on the links \(L_{n,k}\) and \(L'_{n,k}\).

In Sections 5–6–7, we define the rational link surgery complex and obtain a simplified model for two-component L-space links, which we use to derive obstructions to \(L\)-space surgeries. In Section 5, we review preliminaries on Heegaard Floer homology and the related bordered invariants. In Section 6, we review the integer link surgery formula by [34], and use Zemke’s formulation from [35], [44] to prove a link surgery formula that allows rational coefficients; see Theorem 75. In the same section, we show that for two-component \(L\)-space links this complex admits a simplified model; see Theorem 6. In Section 7 we use this simplified model to obtain an obstruction to \(L\)-space surgeries on two-component \(L\)-space links and apply it to \(L_{n,k}.\)

In Sections 8–9, we classify the \(L\)-space surgeries on the links \(L_{n,k}\) and \(L'_{n,k}\). We begin by recalling some properties of the Turaev torsion, and explaining how, by the results of [45], it can be used to determine the set of \(L\)-space fillings of rational homology solid tori. We also discuss, in Section 8.3, the consequences of [45] for manifolds with multiple boundary components, and introduce the notion of weak box-convexity. In Section 9, we focus on the links \(L_{n,k}\) and \(L'_{n,k}\). For \(L_{n,k}\), which we show are \(L\)-space links, we classify their surgeries by combining the arguments developed in Section 55 and Section 7. For the links \(L'_{n,k}\), we use a different technique, based essentially on a detailed study of the Turaev torsions of the manifold obtained by filling only one boundary component of their exteriors.

Finally, there are two remaining sections. In Section 2, we introduce some basic notions on two-bridge links and prove several technical results used throughout the paper. In particular, we use plumbing calculus to describe different diagrams of two-bridge links, which are used in Section 4, and we compute the multivariable Alexander polynomials of \(L_{n,k}\) and \(L'_{n,k}\), which are needed in Sections 7 and Section 9. In Section 10, we prove Theorem 12 on satellite knots.

Acknowledgements↩︎

DS would like to thank Alice Merz for many helpful conversations. HZ would like to thank his postdoc mentor Linh Truong for helpful conversations and generous support. The authors thank Ian Zemke for pointing out a mistake in an earlier draft of this paper and for clarifying some technical details. The authors thank Qingfeng Lyu for pointing out that his result could be used to prove Theorem 4. DS was partially supported by FWF project P 34318 and by ERC Advanced Grant KnotSurf4d. HZ is supported by an AMS-Simons travel grant. The authors thank the Max Planck Institute for Mathematics in Bonn for its hospitality, where this project was initiated.

2 Two-bridge links↩︎

In this section, we recall some basic notions on two-bridge links and prove some technical results that will be needed later in the paper. More precisely, in Section 2.1 we use plumbing calculus to find suitable diagrams of two-bridge links, which will be crucial to construct in Section 4 to construct taut foliations. Next, in Section 2.2, we compute the multivariable Alexander polynomials of the links \(L_{n,k}\) and \(L'_{n,k}\), which we will use in Sections 7.2 and \(\ref{sec:subsec:32class32for32L39nk}\) respectively. We refer to [46] for more details on two-bridge links.

A two-bridge link can be described by a rational number \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers with \(p>0\), \(q\) is odd and \(0<|q|<p\), in the following way. We fix a sequence of integers \((a_1,\dots,a_n)\) such that \[\label{eq:continued32fraction} \frac{p}{q}=a_1-\cfrac{1}{a_2-\cfrac{1}{\ddots-\cfrac{1}{a_n}}}\tag{1}\] and consider the link defined by the diagram (disregarding the orientations for now) in Figure 5. We denote this link by \(L(a_1,\dots, a_n)\)⁴.

We are interested in the case when \(L(a_1,\dots, a_n)\) has two components, and this happens exactly when the fraction \(\frac{p}{q}\) has numerator \(p\) even, see [46]. When \(L(a_1,\dots, a_n)\) is a link we orient the components as in Figure 5.

A priori it could happen that the isotopy class of the two-bridge link associated to \(\frac{p}{q}\) depends on the choice of the continued fraction representation of \(\frac{p}{q}\). This is not the case, by the following theorem.

We denote by \(b(p,q)\) the two-bridge link associated to the rational number \(\frac{p}{q}\). We also recall the following fact, that will be used repeatedly in the paper; see [48].

2.1 Plumbing calculus↩︎

In Section 4, we exploit different diagrammatic presentations of two-bridge links to construct taut foliations on their surgeries. For this purpose, we briefly introduce some notions from plumbing calculus, which provides a convenient way to obtain various diagrams of a two-bridge link. We restrict ourselves to recalling only what is strictly necessary for our purposes and refer to [49] for the general theory of plumbing calculus and to [50] for its connection with links in \(S^3\).

We can encode two-bridge links with linear plumbing graphs, i. e. linear graphs whose vertices have an integer weight assigned to them, by associating to a graph

the (unoriented) two-bridge link \(L(-a_1,-a_2,\dots, -a_n)\), with diagram described in Figure 5.

We introduce the following moves on linear plumbing graphs.

• blow-down: a vertex with weight \(\epsilon\in \{\pm 1\}\) can be removed in the following ways, according to its valence:

  We call blow-up the inverse operation.

• \(0\)-chain absorption: a vertex with weight \(0\) can be removed as shown:

For us, the moves just introduced are relevant for the following fact. We refer to [50] for details.

We collect, for the sake of completeness, some elementary lemmas on linear plumbing graphs that we will need later.

We conclude this section by finding suitable descriptions of two-bridge links. The corresponding diagrams, obtained as in Figure 5, will be used in the next sections to construct taut foliations on their surgeries.

We first address the case of non-fibered two-bridge links with the next two lemmas.

We now turn, with the next lemma, to the fibered case.

2.2 Alexander polynomials↩︎

In this section we compute the multivariable Alexander polynomials of the links \(L_{n,k}\) and \(L'_{n,k}\). These polynomials will be used in Section 7.2 to compute the \(H\)-function of \(L_{n,k}\), and in Section 9.2 to determine the Turaev torsions of the manifolds obtained by filling one component of the links \(L'_{n,k}\). Recall that the multivariable Alexander polynomial \(\Delta_L\) of a link \(L\) with more than one component depends on the orientation of \(L\), so we fix the orientation on the link \(b(p,q)\) as described in Figure 5. Moreover, since the Alexander polynomial is unchanged by taking mirror images, we can assume that \(q\) is positive.

In [52], Hoste gives an explicit formula for \(\Delta_{b(p,q)}\) and describes a useful algorithm to compute it, derived from an interpretation of such formula in terms of walks on the \(2\)-dimensional integer lattice. We note that our conventions for two-bridge links differ from those in [52]: the oriented link \(K_{\frac{p}{q}}\) there corresponds to the oriented link \(b(p,-q)\) in this paper.

The algorithm in [52] proceeds as follows. Let \(\mathcal{A}\) be the set of positive multiples of \(q\) that are strictly less than \(pq\); that is, \[\mathcal{A}=\{iq|\, 0<i<p\}.\] To each multiple \(iq\), associate the quantity \(h_i=(-1)^{\lfloor iq/p\rfloor}\), and, when \(i\neq 1,p-1\), the quantity \(v_i=\frac{h_{i-1}+h_{i+1}}{2}.\) These data determines a path in \(\mathbb{Z}^2\) as follows:

• Start at \((0,0)\in\mathbb{Z}^2\).

• For each even multiple \(2iq\) of \(q\) in \(\mathcal{A}\), take a step from a point \(P\) to the point \(P+(h_{2i},v_{2i})\).

In other words, the horizontal component of the step associated to the multiple \(2iq\) is \(h_{2i}\), while the vertical component depends on the values of \(h_{2i-1}\) and \(h_{2i+1}\): if they are equal and positive is \(+1\), if they are equal and negative is \(-1\), and is \(0\) if they are different. We refer to the vector \((h_{2i}, v_{2i})\) as the coordinates of the step.

In [52], Hoste shows that the multivariable Alexander polynomial of \(b(p,q)\) is represented by the polynomial whose coefficient of \(x^iy^j\) is \((-1)^{i+j}k_{i,j}\), where \(k_{i,j}\) is the number of times the path visits the point \((i,j)\).

2.2.1 Alexander polynomials of the links \(L_{n,k}\).↩︎

We start with the links \(L_{n,k}\), with \(n,k\ge 1\), depicted on the left-hand side of Figure 1. By direct computation, one checks that \[L_{n,k}=b(4nk+2n+2k,-2k-1).\] If we set \(p=2n+1\) and \(q=2k+1\), then \(L_{n,k}=b(p',-q)\), where \(p'=pq-1\). As observed above, in order to compute the Alexander polynomial, we can work with the mirror \(b(p',q)\).

We introduce some notation. As before, let \(\mathcal{A}\) denote the set of multiples of \(q\) contained in \((0,p'q)\), and observe that, since \(p'\) and \(q\) are coprime, \[\mathcal{A}=\mathcal{A}_1\sqcup\cdots\sqcup \mathcal{A}_q,\] where \[\mathcal{A}_m=\mathcal{A}\cap [(m-1)p',mp'], \qquad 1\le m\le q.\] Moreover, since for each fixed \(m\) there are exactly \(p\) multiples of \(q\) in \([(m-1)p',mp']\), the sets \(\mathcal{A}_1\) and \(\mathcal{A}_q\) have cardinality \(|\mathcal{A}_1|=|\mathcal{A}_q|=p-1\), while \(|\mathcal{A}_m|=p\) for \(1<m<q\). Notice that, by definition, if \(iq\in \mathcal{A}_m\), then \(h_i=(-1)^{m-1}\).

In the following proposition, we use the symbol \(\doteq\) to denote equality up to multiplication by units in \(\mathbb{Z}[x^{\pm1},y^{\pm1}]\).

We refer the reader to Figure 44 for (a translate of) the set of lattice points visited by the path in the preceding proof.

2.2.2 Alexander polynomials of the links \(L'_{n,k}\).↩︎

We focus now on the links \(L'_{n,k}\), with \(n,k\ge 1\), described on the right-hand side of Figure 1. They are isotopic to the mirrors of \(b(4nk+2n-2k,2n-1)=b(pq+1,q)\), where \(p=2k+1\) and \(q=2n-1\). For this reason, we work with \(b(pq+1,q)\), and we orient it according to the convention set at the beginning of Section 2, i.e. as in Figure 24. We set \(p'=pq+1\), so that \(L'_{n,k}=b(p',q)\).

As before, we have \[\mathcal{A}=\mathcal{A}_1\sqcup\cdots\sqcup\mathcal{A}_q,\] where \(\mathcal{A}\) is the set of multiples of \(q\) in \((0,p'q)\), and \[\mathcal{A}_m=\mathcal{A}\cap[(m-1)p',mp'], \qquad m=1,\dots,q.\] Each \(\mathcal{A}_m\) has cardinality \(p\), and since \(p\) is odd, if we label the multiples of \(q\) in each \(\mathcal{A}_m\) by \(1,\dots,p\), then the even multiples of \(q\) in \(\mathcal{A}_m\) are those whose parity agrees with that of \(m-1\). Recall that, for \(k\ge 0\), we use the notation \(r_k=1+xy+\cdots+x^k y^k\).

3 Diagrams and persistently foliar links↩︎

The goal of this section is to prove Theorem 30, which is a general theorem regarding the existence of taut foliations on all non-trivial surgeries on \(n\)-component links. This theorem will be applied in Section 4 to two-bridge links.

The section is roughly divided in three parts. We first introduce the notation needed to state Theorem 30. We then recall some notions regarding the theory of branched surfaces, and finally, in Section 3.2, we prove Theorem 30.

Recall that a slope on a torus \(T\) is an element of the projective space \(\mathbb{P}(H_1(T; {\mathbb{R}}))\) and that rational slopes, i.e. the elements of \(\mathbb{P}(H_1(T; {\mathbb{Q}}))\), are in bijection with the set of isotopy classes of (unoriented) essential simple closed curves on \(T\). There is an identification, canonical up to isotopy, between \(T\) and \(\frac{H_1(T;\mathbb{R})}{H_1(T;\mathbb{Z})}\). A linear foliation of slope \(s\) on \(T\) is a foliation isotopic to the image in \(T\) of the foliation by parallel lines of slope \(s\) on \(H_1(T;\mathbb{R})\).

If \(M\) is a compact orientable \(3\)-manifold whose boundary is a union of tori \(T_1,\dots,T_n\), a multislope is an \(n\)-tuple \((s_1,\dots,s_n)\), where each \(s_i\) is a slope on the torus \(T_i\). If \(M\) is the exterior of a link \(L\subset S^3\), we say that a multislope \((s_1,\dots,s_n)\) is non-trivial if none of the \(s_i\) is the meridian of a component of \(L\).

This definition generalises that of a persistently foliar knot, introduced by Delman and Roberts in [29]. Observe that every non-trivial surgery on a persistently foliar link supports a coorientable taut foliation, obtained by capping off the leaves of the taut foliation in the exterior of the link with the meridional discs of the solid tori.

The main theorem of this section states, roughly speaking, that links with special diagrams are persistently foliar. The diagrammatic condition appearing in the statement comes from adapting and extending the main construction of [28] to the case of links with multiple components. We begin by introducing some notation and definitions.

We will always suppose that the diagram is alternating in a twist region. Otherwise, the diagram can be replaced by another one with fewer crossings by using Reidemeister moves.

Suppose that \(D\) is a link diagram whose twist regions all contain at least two crossings. We associate to \(D\) two graphs with \(\mathbb{Z}\)-weighted edges, \(\mathcal{G}_g\) and \(\mathcal{G}_r\), as follows:

• Step 1. We replace each twist region in \(D\) by a blue arc with weight \(w(R)\), as Figure 25–left shows. We denote the resulting graph in \(S^2\) by \(\Gamma\).

• Step 2. We colour the complementary regions of \(\Gamma\) as follows. First, we collapse each blue arc of \(\Gamma\) to a point, obtaining a \(4\)-valent graph \(\Gamma' \subset S^2\), whose complementary regions correspond naturally to those of \(\Gamma\). We then checkerboard-colour the complementary regions of \(\Gamma'\) and transfer this colouring to \(\Gamma\) via the correspondence (see Figure 25–right). For convenience, we colour the regions green and red.

• Step 3. The vertices of the graph \(\mathcal{G}_g\) correspond to the green complementary regions of \(\Gamma\). Each blue arc in \(\Gamma\) that meets some green regions only at its endpoints determines a weighted edge between the corresponding vertices, with weight equal to that of the blue arc. The graph \(\mathcal{G}_r\) is defined analogously, using the red complementary regions and the blue arcs that meet them only at their endpoints.

We refer to Figure 26 for a concrete example of the construction of the graphs.

We denote by \(\mathcal{G}\) the disjoint union \(\mathcal{G}_r\sqcup \mathcal{G}_g\). By construction, the edges of \(\mathcal{G}\) are canonically identified with the twist regions of \(D\). Using this identification, we say that a component of \(L=K_1\sqcup \cdots \sqcup K_n\) passes through an edge of \(\mathcal{G}\) if it passes through the corresponding twist region. We denote by \(\mathcal{R}_i\) the set of edges of \(\mathcal{G}\) the component \(K_i\) passes through.

Figure 26 shows a sample application of the theorem to a link with two components.

The proof of Figure 30 is based on the construction presented in [28]. For this reason, we will often refer the reader to that paper for proofs and more details. At the same time, in order to explain the modifications required to establish the more general statement given here and to clarify as much as possible the key ideas underlying the proof, we need to recall some notions. We begin with branched surfaces

3.1 Branched surfaces↩︎

Let \(M\) be a compact orientable manifold, whose boundary is a possibly empty union of tori.

We define the branch locus of \(B\) as the set of points where \(B\) is not locally homeomorphic to a surface, and the triple points of \(B\) as those where the branch locus is not locally homeomorphic to an arc.

The complement of the branch locus in \(B\) is a union of connected surfaces. The abstract closures of these surfaces – taken under any path metric on \(M\) – are called the sectors of \(B\). Similarly, the complement of the set of triple points within the branch locus is a union of \(1\)-dimensional connected manifolds. To each of these manifolds we associate arrows in \(B\), called cusp directions, indicating the direction of the smoothing, as illustrated in Figure 27.

We denote by \(N_B\) a fibered regular neighbourhood of \(B\), well-defined up to isotopy, whose local structure is depicted in Figure 28. The boundary of \(N_B\) naturally decomposes into three compact subsurfaces: the horizontal boundary \(\partial_h N_B\), the vertical boundary \(\partial_v N_B\), and \(N_B\cap \partial M\). The horizontal boundary is transverse to the interval fibers of \(N_B\), while the vertical boundary intersects them, if at all, in one or two proper closed subintervals contained in their interiors.

3.1.1 From branched surfaces to laminations.↩︎

A common strategy to construct foliations on a \(3\)-manifold is to find intermediate objects, called laminations.

Branched surfaces provide an useful combinatorial tool that can be used to define laminations.

Analogous definitions apply to train tracks on surfaces, and, for later use, we introduce the following notion.

Not all branched surfaces fully carry laminations (see, for example, [53]); however, in [54], Li introduces the notion of laminar branched surface, and shows that they always fully carry laminations. We refer to [54] for details and definitions, and note only that a crucial requirement for a branched surface to be laminar is that it contains no \((\)half\()\) sink discs.

We fix the following notation. Let \(M\) be a manifold whose boundary is union of tori \(T_1, \dots, T_k\), and let \(h\leq k\) be a positive integer. Given a rational multislope \((s_1,\dots, s_h)\), we denote by \(M(s_1,\dots, s_h)\) the manifold obtained by filling the boundary component \(T_i\) along the slope \(s_i\), for \(i=1, \dots, h\). When \(h < k\), this manifold has non-empty boundary.

In the case of our interest, which concern fillings of \(3\)-manifolds with torus boundary, the following theorem plays a key role. We state it in a slightly more general and detailed form than the one appearing in [55]; the additional details follow from its proof. We have not defined the notion of essential lamination, appearing in the statement, since we will not need it; we refer to [53] for details.

3.1.2 From laminations to foliations.↩︎

In general, the presence of a lamination \(\mathcal{L}\) does not guarantee the existence of a foliation extending it. Here, we present one situation in which this does happen, following [29]. We assume some familiarity with sutured manifold theory, and refer the reader to [56] for definitions.

If \(\mathcal{L}\) is a lamination in \(M\) carried by a branched surface \(B\), we will suppose that \(\partial_h N_B\) is contained in \(\mathcal{L}\). Moreover, we denote by \(M_{|\mathcal{L}}\) the metric completion of \(M\setminus \mathcal{L}\) under any path metric on \(M\setminus \mathcal{L}\).

As the name suggests, the noncompact extension property can be used to extend laminations to foliations, as the following result shows.

The statement above differs from that of Delman and Roberts in [29], in that it guarantees the existence of foliations realising multislopes rather than slopes. However, their proof adapts readily to this more general setting, as communicated privately by the authors.

3.2 Proof of Theorem 30↩︎

In this section we prove Theorem 30. Let \(L\subset S^3\) with components \(K_1,\dots, K_n\), and fix a diagram \(D\) of it. Starting from \(D\), we define a new link \(\mathcal{L}\subset S^3\) as follows. For each twist region, we first remove all crossings in it if it contains an even number of them, and all but one otherwise. Then, we encircle the two strands with an unknotted circle, called a crossing circle. The link \(\mathcal{L}\) has \(n\) components, that we denote \(K'_1,\dots, K'_n\) corresponding to those of \(L\), and additional components \(J_1,\dots, J_m\) given by the crossing circles, one for each twist region of \(D\). Each crossing circle bounds a twice-punctured disc in the exterior of \(\mathcal{L}\), that we call crossing disc, see Figure 29.

By construction, the link \(L\) is the image of \(K'_1 \sqcup \cdots \sqcup K'_n\) after an appropriate Dehn surgery along the crossing circles. More precisely, if the twist region associated to the crossing circle \(J_i\) contains \(2k_i\) or \(2k_i+1\) left-handed crossings, with \(k_i\geq 0\), the surgery coefficient of \(J_i\) is \(\frac{1}{k_i}\). Analogously, if the twist region contains \(-2k_i\) or \(-(2k_i-1)\) right-handed crossings, with \(k_i\leq 0\), the surgery coefficient is \(\frac{1}{k_i}\).

The overall strategy of the proof is the following. We aim to construct a branched surface in the exterior of \(\mathcal{L}\), and use it, together with Theorems 36 and 39, to construct taut foliations in the exterior of \(\mathcal{L}\) intersecting:

• the boundary components corresponding to the crossing circles in linear foliations of multislope \((\frac{1}{k_1},\dots, \frac{1}{k_m})\);

• the boundary components corresponding to the remaining components of \(\mathcal{L}\) in linear foliation of any non-trivial multislope \((s_1,\dots, s_n)\).

Such a situation would imply that \(L\) is persistently foliar, proving the theorem.

Let \(M\) denote the exterior of \(\mathcal{L}\), with \(\partial_i M\) being the boundary component corresponding to \(K'_i\), for \(i=1,\dots n\). For each of these boundary components, we fix a boundary parallel torus \(T_i=\partial_i M\times \{1\}\), where \[\nu\partial_i M=\partial_i M \times [0,1], \text{ with } \partial_i M= \partial_i M\times \{0\}\] is a fixed collar of \(\partial_i M\).

Our first step is to define a \(2\)-complex \(\Sigma\) in \(M\), that we will later smooth into a branched surface. The \(2\)-complex \(\Sigma\) is defined as the union of:

• The boundary parallel tori \(T_1,\dots, T_n\).

• The closures of the intersections between the crossing discs \(D_1,\dots, D_m\), bounded by the components \(J_1,\dots, J_m\) respectively, and \(M\setminus(\nu\partial_1 M\cup\cdots \cup \nu\partial_n M)\). By a slight abuse of notation, we still denote this discs \(D_1,\dots, D_m\) and refer to them as the crossing discs.

• The closures in \(M\) of the connected components of \[S^2\setminus (D_1\cup\cdots \cup D_m \cup \nu\partial_1 M\cup\cdots \cup \nu\partial_n M)\] when the weights of all twist regions in \(D\) are even; otherwise, we proceed as follows: consider the image of \[D_1\cup\cdots \cup D_m \cup \nu\partial_1 M\cup\cdots \cup \nu\partial_n M\] in the exterior of \(\mathcal{L}'\) after the mutation, take the closures of the connected components of its complement in \(S^2\), and then take their preimages in \(M\) under the mutation. In both cases, we denote by \(\mathcal{S}\) the collection of surfaces obtained in this way.

Notice that each element of \(S\) is an immersed disc in \(M\) whose interior is embedded. Figure 30 shows an example of \(\Sigma\) in the case when all twist regions have an even number of crossings. The exterior of \(\Sigma\) has a very simple structure, as shown by the following lemma, proved in [28].

We now assign coorientations to the surfaces composing \(\Sigma\), which we will use to determine a smoothing of \(\Sigma\). We denote by \(\delta_1 M,\dots, \delta_m M\) the boundary components of \(M\) corresponding to the crossing circles \(J_1,\dots J_m\). Observe that \(\delta_1 M,\dots, \delta_m M\) intersect \(\Sigma\) transversely, while the remaining boundary components are disjoint from \(\Sigma\). We want the branched surface we obtain from \(\Sigma\) to have the following two important properties: the boundary train track induced on \(\delta_h M\) must realise the slope \(\frac{1}{k_h}\) for \(h=1,\dots, m\), and each torus \(T_i\) must have two cusps, for \(i=1,\dots, n\).

Figure 31 shows an example of a cusp. Note that once a coorientation of the crossing disc \(D\) intersecting \(T\) along \(c\) is chosen, there is a unique choice of coorientation of \(T\setminus c\) in a neighbourhood of \(c\) such that smoothing according to these coorientations produces a cusp.

Before starting to assign coorientations, we make explicit a strong relation between the combinatorial properties of the diagram of the link \(\mathcal{L}\) and the graphs associated to the diagram \(D\) of \(L\) at the beginning of Section 3. From now on we suppose that the diagram \(D\) of \(L\) satisfies the hypotheses of Theorem 30. In particular, all twist regions contain at least two crossings, so that the graphs \(\Gamma\), \(\mathcal{G}_r\), \(\mathcal{G}_g\) and \(\mathcal{G}\) can be defined. By construction there is a canonical bijection between \(\mathcal{S}\) and the set of connected components of \(S^2\setminus \Gamma\), and hence to the vertices of \(\mathcal{G}\). More precisely, the following properties hold:

• each element of \(\mathcal{S}\) corresponds bijectively to a vertex of \(\mathcal{G}\);

• every crossing disc corresponds bijectively to an edge of \(\mathcal{G}\);

• an element in \(\mathcal{S}\) intersects the boundary of a crossing disc \(D\) if and only if the corresponding vertex in \(\mathcal{G}\) is adjacent to the edge corresponding to \(D\).

By using this dictionary, we have that a component \(K_i\) of \(L\) passes through an edge of \(\mathcal{G}\) if and only if the corresponding component \(K_i'\) of \(\mathcal{L}\) intersects the crossing disc associated to that edge. Moreover, we can define the weight \(w(D)\) of a crossing disc \(D\) as the weight of the corresponding twist region (and edge in \(\mathcal{G}\)). The following lemma is simply a restatement in this setting of some of the hypotheses of Theorem 30.

3.2.1 Coorientations of the elements in \(\mathcal{S}\).↩︎

We fix a colouring of the connected components of \(S^2\setminus \Gamma\), and use it, together with the bijection mentioned in the previous section, to assign coorientations to the elements in \(\mathcal{S}\). More precisely, recall that \(X_1\) and \(X_2\) denote the two balls in the exterior of \(\Sigma\). For consistency, we adopt the convention that our figures are drawn from the viewpoint of a reader located inside the ball \(X_1\). We impose the coorientation to point into \(X_1\) (resp. into \(X_2\)) for sectors corresponding to green (resp. red) complementary regions of \(\Gamma\). We will also adhere to the convention that the colour green (resp. red) indicates the positive (resp. negative) side of a sector of \(\Sigma\). See Figure 32 for examples of this coorientation near a crossing disc, both when the disc is adjacent to a crossing and when it is not.

3.2.2 Coorientations of the crossing discs and the boundary parallel tori.↩︎

Let us fix, for each \(i=1,\dots, n\), the crossing discs \(D^{1,2}_i\) given by Lemma 42. Moreover, for each \(i\), we fix circles of intersections \[c^1_i\subset T_i\cap D_i^1 \text{ and } c^2_i\subset T_i\cap D_i^2.\]

Figure 33 shows an example in the case the twist region has weight \(2\).

In the next lemma, we prescribe coorientations for the boundary parallel tori and the discs \(D_i^j\).

From Lemma 44 we know how to obtain the desired two cusps on each boundary parallel torus. The goal now is to show that we can coorient the remaining crossing discs and finally smooth \(\Sigma\) so that the boundary train track of the resulting branched surface on \(\delta_h M\) realises the slope \(\frac{1}{k_h}\), for \(h=1,\dots, m\).

More precisely, we are interested in smoothing a subcomplex of \(\Sigma\). Let \(S\) be the element of \(\mathcal{S}\) given by Lemma 42, and denote by \(\mathcal{D}_S\) the set of crossing discs whose boundary intersects \(S\). Notice that, by hypothesis, none of the \(D_i^j\) belongs to \(\mathcal{D}_S\). Also notice, that since \(S\) is cooriented (and oriented, using the ambient orientation), its intersection with the boundary parallel tori \(T_i\), that we denote by \(\sigma\), is naturally oriented.

For example, the disc in Figure 33 is \(S\)-cooriented if \(S\) is the region on the right of the picture.

In the next section we will need a slight variation of the previous lemma. In fact, in the same way, using the arguments of [28], one proves the following.

Examples where this addendum is useful appear in Section 4; see Figures 35, 36, 37 and 38.

The following properties of the branched surface \(B\) are proven by applying verbatim the arguments in [28], to which we refer the reader. We only point out that here is where the hypothesis that the graphs \(\mathcal{G}_r\) and \(\mathcal{G}_g\) are connected is used, since it plays a crucial role in proving that \(X\) is a product sutured ball.

We are finally ready to prove Theorem 30.

4 Persistently foliar two-bridge links↩︎

The main goal of this section is to apply Theorem 30 to the study of two-bridge links, and prove the following result. Recall the links \(L_{n,k}\), \(L'_{n,k}\) defined in Figure 1.

To prove Theorem 50, we first consider the non-fibered two-bridge links satisfying the hypotheses of the statement, and then the fibered ones. We remark that in [32] it is shown that all non-trivial surgeries on a fibered two-bridge link satisfying the hypotheses of Theorem 50 support coorientable taut foliations. In this case, Theorem 50, besides providing a different proof, covers also the case of non-rational multislopes.

In Section 4.3, that is the last part of this section, we construct taut foliations on some surgeries on the links \(L_{n,k}\) and \(L'_{n,k}\). The results proved there will be used both in the classification of their \(L\)-space surgeries in Section 9 and in the applications to satellite knots in Section 10. Although the proofs in Section 4.3 also use branched surfaces, they rely on a different construction – natural for fibered links – from the one presented in the previous section. Such a construction has been used quite often in the literature; for this reason, we will not give full details, but provide references.

We fix some notations. Consider a two-bridge link \(L=L(a_1,\dots, a_m)\), denote its component by \(K_1\) and \(K_2\), and let \(D\) be the diagram associated to it as in Figure 5. As in the proof of Lemma 21, we can always suppose that \(|a_i|\geq 2\) for each \(i=1,\dots, m\). We can hence consider the graphs \(\mathcal{G}_r\), \(\mathcal{G}_g\) and \(\mathcal{G}=\mathcal{G}_r\cup \mathcal{G}_g\) associated to \(D\). Notice that, by definition, the diagram \(D\) has a twist region containing \(|a_j|\) crossings, for each \(j=1,\dots, m\). We denote this twist region and – with slight abuse of notation – the corresponding edge in \(\mathcal{G}\) by \(R_j\).

4.1 When \(L\) is non-fibered↩︎

Let \(L\) be a two-bridge link, and assume that it is not isotopic to \(L_{n,k}, L'_{n,k}\) or their mirrors. By Remark 2, this is equivalent to saying that \(L\) is not isotopic to any link \(L(2,\dots, 2,2z)\), with \(z\in\mathbb{Z}\). If in addition \(L\) is non-fibered, then we can write \(L=L(a_1,\dots, a_m)\) as described in the statement of Lemma 21. Since \(a_j\) is even for each odd \(j\), one of the two components of \(L\), say \(K_1\), has the property that \(K_1\) passes through a twist region \(R_j\) if and only if \(j\) is odd. In particular, every edge in \(\mathcal{R}_1\) has even weight. We also point out that the component \(K_2\) passes through all twist regions of \(D\).

With the next proposition, we cover the case \(m=3\).

4.2 When \(L\) is fibered↩︎

We conclude the proof of Theorem 50 by studying the case when \(L\) is fibered.

4.3 Taut foliations and the links \(L_{n,k}\) and \(L'_{n,k}\)↩︎

We conclude the section by construction taut foliations on some of the manifolds obtained by Dehn surgeries along the links \(L_{n,k}\) and \(L'_{n,k}\) of Figure 1. We will use these results in Sections 9 an 10.

The statement of Proposition 55 is not optimal, as it does not provide taut foliations on all surgeries along \(L_{n,k}\) that are not L-spaces. However, it is enough to obtain interesting applications to satellite knots, see Section 10. We will improve Proposition 55 in Section 9.1, where, by combining it with a recent theorem of Lyu [33] and Theorem 3, we construct taut foliations on all non-\(L\)-space surgeries on the links \(L_{n,k}\); see Theorem 4

In the next proposition, we focus on the links \(L'_{n,k}\).

5 Preliminaries on Heegaard Floer homology↩︎

In this section, we briefly review some preliminaries on Heegaard Floer homology needed for Sections 6 and 7. In Section 5.1, we recall the definitions of Heegaard Floer homology and the link Floer complex; then, in Section 5.2, we review the definitions of type-\(A\) and type-\(D\) modules and bimodules following [61], [62] and in Section 5.3 and 5.4 we review the definitions of the link surgery algebra and the type-\(A\) module for solid tori from [35].

5.1 Heegaard Floer homology and the link Floer complex↩︎

In [63], Ozsváth-Szabó defined a topological invariant for a three manifold \(M\), the Heegaard Floer chain complex, which comes in different flavors. The simplest version is the hat flavor \(\widehat{\mathit{CF}}(M)\), which takes the form of a chain complex over $ {}= {}/2 {}\(. The *minus flavor*\)^-(M)$ takes the form of a free chain complex over the ring $ {}[U]$, where \(U\) is a formal variable. We will also consider the algebraic completion of \(\mathit{CF}^-(M)\) over $ {}[U],$ denoted \(\boldsymbol{\mathbf{CF}}^{-}(M)\), which is a chain complex over the power series ring $ {}U .$ Setting \(U=0\) in \(\mathit{CF}^-(M)\) or \(\boldsymbol{\mathbf{CF}}^{-}(M)\) recovers \(\widehat{\mathit{CF}}(M)\). Taking the homology yields the Heegaard Floer homologies \(\widehat{\mathit{HF}}(M)\), \(\mathit{HF}^-(M)\) and \(\boldsymbol{\mathbf{HF}}^{-}(M)\), which take the form of an $ {}$-vector space, and modules over $ {}[U]$ and $ {}U ,$ respectively. The Heegaard Floer chain complex and homology admit a homological $ {}$-grading, denoted \(gr\), such that \(gr(U)=-2\) and \(gr(\partial)=-1\).

The Heegaard Floer chain complex and homology decompose over the set spin\(^c\) structures Spin\(^c(M)\), for which there is a non-canonical identification Spin\(^c(M) \cong H_1(M; {\mathbb{Z}})\). We have \[\widehat\mathit{CF}(M) = \bigoplus_{\boldsymbol{t}\in \operatorname{Spin}^c(M)} \widehat\mathit{CF}(M,\boldsymbol{t}).\] The same decomposition holds for other flavors and for the homology as well. Ozsváth-Szabó proved in [64] that for a rational homology sphere \(M\) it holds \(\operatorname{dim}\widehat\mathit{HF}(M,\boldsymbol{t}) \geq 1\) for each \(\boldsymbol{t}\in \operatorname{Spin}^c(M).\)

In [65], Ozsváth-Szabó defined a refinement of the Heegaard Floer chain complex, which is an invariant of a link \(L \subset M\). Later, Zemke provided a reformulation of this invariant. In the following we recall some of its important properties. For a detailed exposition, the reader is referred to [66].

Given an \(n\)-component link \(L=K_1\cup \dots \cup K_n\) in a rational homology sphere \(M\), the link Floer complex \(\mathcal{C\!F\!L}(L)\) is a finitely generated free chain complex over the ring $ {}[W_1,Z_1,,W_n,Z_n]$. If we set \(U_i=W_iZ_i\), then the homological actions induced by multiplication by \(U_i\) coincide for all \(i=1,\dots, n\). Define the link lattice by \[ {\mathbb{H}}(L) := \prod_{i=1}^n \left( {\mathbb{Z}}+ \frac{\operatorname{lk}(L,L-K_i)}{2} \right).\] Each generator of \(\mathcal{C\!F\!L}(L)\) has an Alexander grading \(\boldsymbol{s}= (s_1,\dots,s_n) \in {\mathbb{H}}\) and the algebraic elements interact with the Alexander grading as follows: for \(i=1,\dots,n,\) multiplication by \(Z_i\) increases \(s_i\) by \(1\) and multiplication by \(W_i\) decreases \(s_i\) by \(1\). In particular, \(U_i\) preserves the Alexander grading. The differential preserves Alexander grading, and there is a decomposition as a chain complex \[\mathcal{C\!F\!L}(L) = \bigoplus_{\boldsymbol{s}\in{\mathbb{H}}(L)} \mathcal{C\!F\!L}(L,\boldsymbol{s}),\] where each \(\mathcal{C\!F\!L}(L,\boldsymbol{s})\) is a chain complex over $ {}[U_1,,U_n].$ Specifically, suppose \(\{\boldsymbol{\mathbf{y}}_1,\dots,\boldsymbol{\mathbf{y}}_m\}\) is a free basis of \(\mathcal{C\!F\!L}(L)\). Then each \(\mathcal{C\!F\!L}(L,\boldsymbol{s})\) is freely generated over $ {}[U_1,,U_n]$ by \(\{a_1 \boldsymbol{\mathbf{y}}_1,\dots, a_m \boldsymbol{\mathbf{y}}_m \}\), where \(a_i \in {\mathbb{F}}[W_1,Z_1,\dots,W_n,Z_n]\) is a monomial.

There is a homological grading on \(\mathcal{C\!F\!L}(L)\), where \(gr(W_i)=-2, gr(Z_i)=0\) and \(gr(\partial)=-1\).

We recall from the introduction the definition of an \(L\)-space link by [15].

5.2 Type-\(A\) and type-\(D\) modules and bimodules↩︎

We recall several definitions regarding bordered modules. All definitions in this section are taken from [61], [62]. We always assume the ground ring \(\boldsymbol{r}\) has characteristic \(2.\)

Once we have a type-\(D\) module, we can define maps \(\delta^i\) for \(i\geq 0\) by setting \[\delta^0 = \mathbb{I}_A, \qquad \delta^i = (\delta^1 \otimes \mathbb{I}_{A^{\otimes i-1}} ) \circ \delta^{i-1} \qquad i > 0.\] In the case of our interest, type-\(D\) modules will be bounded; see, [61]. This means that for each \(x\in N\), there exists an here is an integer \(m\) so that for all \(i\geq m\), \(\delta^i(x)=0\). In this case, we set \[\delta := \sum_{i=0}^\infty \delta^i: N \to N \otimes_{\boldsymbol{r}} \mathcal{T}^*(A),\] where \[\mathcal{T}^*(A) := \bigoplus_{n=0}^{\infty} A^{\otimes n}.\]

If \(A\) is commutative, a type-\(A\) module \({}_{A}\mathcal{M}\) with \(m_i = 0\) for \(i>2\) is a regular chain complex over the ring \(A\), with the differential given by \(m_1\) and multiplication by elements in \(A\) given by \(m_2\).

The notion of a type-\(A\) module can be generalized to define a bimodule.

Given a right type-\(D\) module \(\mathcal{N}^A\) and a type-\(AA\) bimodule \({}_A\mathcal{M}_B\), the box tensor product \(\mathcal{N}^A \boxtimes {}_A\mathcal{M}_B\) is a right type-\(A\) module over \(B\), with underlying module \(N\otimes_{\boldsymbol{r}} M\) and structure maps given by the following diagram \[\begin{tikzcd}[column sep=.5cm,row sep=.4cm] \ar[d, dashed] & \ar[dashed,dd] & \ar[ddl, Rightarrow]\\ \delta \ar[dd, dashed] \ar[dr, Rightarrow] && \\ & m \ar[dashed,d]&.\\ \;&\;& \end{tikzcd}\]

5.3 Knot and link surgery algebras↩︎

In this section we recall the definitions of the knot and link surgery algebras, following [35]. Before doing so, we premise a brief discussion regarding algebras on idempotent rings.

Assume that \(A\) is an associative algebra over the idempotent ring \(\boldsymbol{\mathbf{I}}=\boldsymbol{\mathbf{I}}_0 \oplus \boldsymbol{\mathbf{I}}_1\). Here, each \(\boldsymbol{\mathbf{I}}_i\) is a copy of $ {}= {}/2 {}$, and we denote by \(i_0\) and \(i_1\) the generators of \(\boldsymbol{\mathbf{I}}_0\) and \(\boldsymbol{\mathbf{I}}_1\) respectively. Then, as a module over \(\boldsymbol{\mathbf{I}}\), we can decompose \(A\) as \[A=\left( \boldsymbol{\mathbf{I}}_0 \cdot A \cdot \boldsymbol{\mathbf{I}}_0\right) \oplus \left(\boldsymbol{\mathbf{I}}_0 \cdot A \cdot \boldsymbol{\mathbf{I}}_1\right) \oplus \left(\boldsymbol{\mathbf{I}}_1 \cdot A \cdot \boldsymbol{\mathbf{I}}_0\right) \oplus \left(\boldsymbol{\mathbf{I}}_1 \cdot A \cdot \boldsymbol{\mathbf{I}}_1\right)\] In fact, it is not difficult to see that the above subalgebras have pairwise trivial intersections, and they generate since for every \(a\in A\) we have: \[a=(i_0+i_1)a(i_0+i_1)=(i_0 a i_0) + (i_0 a i_1) + (i_1 a i_0) + (i_1 a i_1).\] By reversing this reasoning, we see that in order to define an algebra over \(\boldsymbol{\mathbf{I}}\) it is enough to describe the four submodules above, and the product rules among their elements.

We call \(\boldsymbol{\mathbf{I}}_0 \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_0\) and \(\boldsymbol{\mathbf{I}}_1 \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_1\) the idempotent \(0\) and \(1\) of \(\mathcal{K}\), respectively. Observe that they satisfy the property \((\boldsymbol{\mathbf{I}}_\varepsilon \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_\varepsilon) \cdot (\boldsymbol{\mathbf{I}}_\varepsilon \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_\varepsilon) = \boldsymbol{\mathbf{I}}_\varepsilon \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\) for \(\varepsilon =0,1.\)

We give an explicit description of the link surgery algebra as follows. Define \(\mathbb{E}_n := \{ 0,1 \}^n\). For \(\varepsilon,\varepsilon' \in \mathbb{E}_n\), we write \(\varepsilon\leq \varepsilon'\) if the inequality holds in each coordinate. Similarly, we write \(\varepsilon< \varepsilon'\) if \(\varepsilon\leq \varepsilon'\) and \(\varepsilon\ne \varepsilon'\). We begin by describing the subalgebras \(\boldsymbol{\mathbf{I}}_{\varepsilon'} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\), for \(\varepsilon, \varepsilon'\in \mathbb{E}_n\), and then the products between their elements.

There are \(2^n\) idempotents in \(\mathcal{L}\), namely \(\boldsymbol{\mathbf{I}}_\varepsilon\cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\) for each \(\varepsilon= (\varepsilon_1, \dots, \varepsilon_n) \in \mathbb{E}_n\), where \(\boldsymbol{\mathbf{I}}_\varepsilon:= \otimes_i^n\boldsymbol{\mathbf{I}}_{\varepsilon_i}\). More concretely, \(\boldsymbol{\mathbf{I}}_\varepsilon\cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\) is the tensor of $ {}[Z_i,W_i]$ for those indices \(i\) such that \(\varepsilon_i =0\) and $ {}[U_i, T_i,T^{-1}_i]$ for the indices \(i\) such that \(\varepsilon_i =1\). Given \(\varepsilon,\varepsilon' \in \mathbb{E}_n,\) if \(\varepsilon\leq \varepsilon'\), then \(\boldsymbol{\mathbf{I}}_{\varepsilon'} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\) is the tensor of $ {}[U_i, T_i,T^{-1}_i]_i, _i $ for \(i\) such that \(\varepsilon'_i-\varepsilon_i=1\) and \(\boldsymbol{\mathbf{I}}_{\varepsilon_i} \cdot \mathcal{K}_i \cdot \boldsymbol{\mathbf{I}}_{\varepsilon_i}\) for \(i\) such that \(\varepsilon'_i-\varepsilon_i=0\); otherwise \(\boldsymbol{\mathbf{I}}_{\varepsilon'} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon= 0\).

Concerning products between elements of the above subalgebras, the only ones that require some discussion are those between elements of \(\boldsymbol{\mathbf{I}}_{\varepsilon'} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\) and \(\boldsymbol{\mathbf{I}}_{\varepsilon} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\), when \(\varepsilon< \varepsilon'\). We fix the following notation. Given \(\varepsilon< \varepsilon'\), suppose that \(t\in \boldsymbol{\mathbf{I}}_{\varepsilon'} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\) is in the form of \(t = \otimes_i^n t_i\), where \(t_i = \sigma_i\) or \(\tau_i\) for the indices \(i\) with \(\varepsilon_i'-\varepsilon_i=1\) and \(t_i = 1\) for those with \(\varepsilon_i'-\varepsilon_i=0\). Define \[g^t(a_1 \otimes\cdots \otimes a_n) = \otimes_i^n g^{t_i}(a_i)\] where \(g^{\sigma_i}\) and \(g^{\tau_i}\) are defined as before, and \(g^{t_i}\) is the identity map if \(t_i=1\). Then the products between elements in \(\boldsymbol{\mathbf{I}}_{\varepsilon'} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\) and \(\boldsymbol{\mathbf{I}}_{\varepsilon} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_\varepsilon\) are obtained by imposing \[t \cdot a = g^t(a) \cdot t\] for any \(a\in \boldsymbol{\mathbf{I}}_{\varepsilon}\cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_{\varepsilon}.\)

The link surgery algebra is used to define a type-\(D\) module \(\mathcal{X}(K)^{\mathcal{L}}\) related to the link surgery formula, which we will describe in Section 6.3.

5.4 Type-\(A\) modules for solid tori↩︎

Another component of the link surgery formula from the bordered module perspective is the bordered modules for a disjoint union of solid tori, which we now discuss.

We start by describing the type-\(A\) module \({}_{\mathcal{K}}\mathcal{D}_\lambda\) for a solid torus with integral framing \(\lambda\), defined in [35]. Since \({}_{\mathcal{K}}\mathcal{D}_\lambda\) is a module over \(\boldsymbol{\mathbf{I}}\), arguing as in the beginning of Section 5.3, we have that it is enough to describe the submodules \(\boldsymbol{\mathbf{I}}_0 \cdot \mathcal{D}_\lambda\) and \(\boldsymbol{\mathbf{I}}_1 \cdot \mathcal{D}_\lambda\). The underlying space \(\boldsymbol{\mathbf{I}}_0 \cdot \mathcal{D}_\lambda\) is freely generated over $ {}[W,Z]$ by a generator \(\boldsymbol{\mathbf{x}}^0\) and \(\boldsymbol{\mathbf{I}}_1 \cdot \mathcal{D}_\lambda\) is freely generated over $ {}[U,T,T^{-1}]$ by a generator \(\boldsymbol{\mathbf{x}}^1\). All structure maps \(m_i\) vanish for \(i\neq 2\) and the maps \(m_2\) are defined as follows. If \(a\in \boldsymbol{\mathbf{I}}_i \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_i\) and \(x\in \boldsymbol{\mathbf{I}}_j \cdot \mathcal{D}_\lambda\), then we define \(m_2(a,x)=a\cdot x\) if \(i=j\), and to be \(0\) otherwise. We define \[m_2(\sigma, a\boldsymbol{\mathbf{x}}^0) = g^\sigma (a) \boldsymbol{\mathbf{x}}^1 \qquad m_2(\tau, a\boldsymbol{\mathbf{x}}^0) = g^\tau (a) \boldsymbol{\mathbf{x}}^1\] for \(a \in {\mathbb{F}}[W,Z].\)

The module \({}_{\mathcal{K}}\mathcal{D}_\lambda\) can be viewed as a type-\(AA\) bimodule \({}_{\mathcal{K}}[\mathcal{D}_\lambda]_{ {\mathbb{F}}[U]}\), where \(m_{0,1,1}\) is defined by the multiplication by power of \(U\) and and \(m_{1,1,0}\) is the same as \(m_2\) on \({}_{\mathcal{K}}\mathcal{D}_\lambda\). All other structure maps \(m_{i,1,j}\) are set to be zero.

Generalising this to the setting of rationally framed solid torus, the type-\(D\) module \(\mathcal{D}^\mathcal{K}_{p/q}\) is defined in [35]. For \(\nu \in \{0,1\}\), the underlying space of \(\mathcal{D}^\mathcal{K}_{p/q}\cdot \boldsymbol{\mathbf{I}}_{\nu}\) is the $ {}\(-span of\) }$ and the type-\(D\) structure map is given by \[\delta^1(\boldsymbol{\mathbf{x}}^0_k) = \boldsymbol{\mathbf{x}}^1_k \otimes \sigma + \boldsymbol{\mathbf{x}}^1_{k+p} \otimes T^{\floor{\frac{k+p}{q}}} \tau\] where we choose the representative of \(k\in {\mathbb{Z}}/q {\mathbb{Z}}\) satisfying \(0\leq k<q\).

We can obtain the type-\(A\) module \({}_{\mathcal{K}}\mathcal{D}_{p/q}\) by \({}_{\mathcal{K}}\mathcal{D}_{p/q} := \mathcal{D}^\mathcal{K}_{p/q} \boxtimes {}_{\mathcal{K}| \mathcal{K}} [\mathbb{I}^{\Supset}]\) where \({}_{\mathcal{K}| \mathcal{K}} [\mathbb{I}^{\Supset}]\) is the identity bimodule defined in [35]. We now give an explicit description of \({}_{\mathcal{K}}\mathcal{D}_{p/q}\).

Suppose that \(p\in {\mathbb{Z}},q\in {\mathbb{Z}}_{>0}\) and that \(p\) and \(q\) are coprime. The underlying space \(\boldsymbol{\mathbf{I}}_0 \cdot \mathcal{D}_{p/q}\) is freely generated over $ {}[W,Z]$ by \(q\) generators \(\{\boldsymbol{\mathbf{x}}^0_0, \dots,\boldsymbol{\mathbf{x}}^0_{q-1}\}\) and \(\boldsymbol{\mathbf{I}}_1 \cdot \mathcal{D}_{p/q}\) is freely generated over $ {}[U,T,T^{-1}]$ by \(q\) generators \(\{\boldsymbol{\mathbf{x}}^1_0,\dots,\boldsymbol{\mathbf{x}}^1_{q-1}\}\). We view the generators \(\boldsymbol{\mathbf{x}}^i_k\) as indexed by \(i\in \{0,1\}\) and \(k\in {\mathbb{Z}}/q {\mathbb{Z}}.\)

As before, all structure maps \(m_i\) vanish for \(i\neq 2\) and the maps \(m_2\) are defined as follows. If \(a\in \boldsymbol{\mathbf{I}}_i \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_i\) and \(x\in \boldsymbol{\mathbf{I}}_j \cdot \mathcal{D}_\lambda\), define \(m_2(a,x)= a \cdot x\) if \(i=j\) and to be zero otherwise. We define \[m_2(\sigma, a\boldsymbol{\mathbf{x}}^0_k) = g^\sigma (a) \cdot \boldsymbol{\mathbf{x}}^1_k \qquad m_2(\tau, a\boldsymbol{\mathbf{x}}^0_k) = T^{\floor{\frac{k+p}{q}}}g^\tau (a) \cdot \boldsymbol{\mathbf{x}}^1_{k+p}\] for \(k\in {\mathbb{Z}}/q {\mathbb{Z}}, a \in {\mathbb{F}}[W,Z].\)

The following is a schematic drawing of \({}_{\mathcal{K}}\mathcal{D}_{1/3}\), where an arrow from \(\boldsymbol{\mathbf{x}}\) to \(\boldsymbol{\mathbf{y}}\) labelled by \(u|a\) means that \(m_2(u,\boldsymbol{\mathbf{x}})=a\otimes \boldsymbol{\mathbf{y}}.\) \[\begin{tikzcd} [labels=description, column sep=1.4cm] \boldsymbol{\mathbf{x}}^0_0 \arrow[d, "\sigma|1" description, crossing over] & \boldsymbol{\mathbf{x}}^0_1 & \boldsymbol{\mathbf{x}}^0_2 \arrow[d, "\sigma|1" description, crossing over] \arrow[dll, "\tau|T" description, pos=0.15] \\[10 mm] \boldsymbol{\mathbf{x}}^1_0 & \arrow[from=ul, "\tau|1" description, crossing over] \arrow[from=u, "\sigma|1" description, crossing over] \boldsymbol{\mathbf{x}}^1_1 & \arrow[from=ul, "\tau|1" description, crossing over] \boldsymbol{\mathbf{x}}^1_2 \end{tikzcd}\]

We are now ready to define the type-\(A\) module of the disjoint union of \(n\) (rationally framed) solid tori \({}_{\mathcal{L}}\mathcal{D}_\Lambda\).

As in the case of one solid torus, \({}_{\mathcal{L}}\mathcal{D}_\Lambda\) can be viewed as a type-\(AA\) bimodule \({}_{\mathcal{L}}[\mathcal{D}_\Lambda]_{ {\mathbb{F}}[U_1,\dots,U_n]}\), where \(m_{0,1,1}\) is multiplications by \(U_i\), \(m_{1,1,0}\) is the same as \(m_2\) on \({}_{\mathcal{L}}\mathcal{D}_\Lambda\) and \(m_{i,1,j}=0\) for all other \(i\) and \(j\).

6 Rational link surgery formula↩︎

In this section we deduce a rational link surgery formula as a direct consequence of the works by Zemke [35], [44]. Then we use Liu’s work [21] to show that the rational link surgery complex admits a simplified model in the case of two-component \(L\)-space links. In Section 6.1-6.3 we review the statement of the integral link surgery formula in [34], the definition of the \(H\)-function (in the case of \(L\)-space links) by [37] and Zemke’s reformulation of the link surgery formula using bordered modules [35]. In Section 6.4, we define the rational link surgery complex and prove that its homology is the same the Heegaard Floer homology of the corresponding rational surgery using the works by Zemke. Finally, in Section 6.5, we obtain a simplified model of the link surgery complex by directly applying Liu’s results.

6.1 The integral link surgery formula↩︎

We recall the construction of the integral link surgery complex by Manolescu-Ozsváth [34], following a slight reformulation by Zemke [35]. Before doing so, we introduce the notion of hypercube of chain complexes from [34]. This formalism appears naturally in this context, and we will make use of it in what follows. Recall the notation \(\mathbb{E}_n = \{0,1 \}^n\).

Suppose \(L=K_1 \cup \cdots \cup K_n \subset S^3\) is an oriented \(n\)-component link. Let \(\Lambda =(\lambda_1, \dots, \lambda_n) \in {\mathbb{Z}}^n\) be a set of integral surgery framings. We often conflate it with the surgery matrix \(\Lambda = ( \Lambda_{i,j})_{n\times n}\) defined by \[\begin{align} \Lambda_{i,j} = \begin{cases} \lambda_i &i=j \\ \operatorname{lk}(K_i,K_j) \qquad & i\neq j. \end{cases} \end{align}\]

Given \(\varepsilon= (\varepsilon_1, \dots, \varepsilon_n) \in \mathbb{E}_n = \{0,1 \}^n,\) denote by \(S_\varepsilon\) the multiplicatively closed set in $ {}[W_1,Z_1,,W_n,Z_n]$ generated by \(Z_i\) for \(i\) such that \(\varepsilon_i = 1\). Let \(S_\varepsilon^{-1}\mathcal{C\!F\!L}(L)\) be the localisation of \(\mathcal{C\!F\!L}(L)\) by \(S_\varepsilon\). There is a decomposition \[S_\varepsilon^{-1}\mathcal{C\!F\!L}(L) = \bigoplus_{\boldsymbol{s}\in{\mathbb{H}}(L)} S_\varepsilon^{-1}\mathcal{C\!F\!L}(L,\boldsymbol{s})\] coming from the Spin\(^c\) decomposition of \(\mathcal{C\!F\!L}(L)\), where each \(S_\varepsilon^{-1}\mathcal{C\!F\!L}(L,\boldsymbol{s})\) is a free complex over $ {}[ U_1,,U_n ].$ We define \(C_\varepsilon(\boldsymbol{s})\) to be the free complex over the power series ring $ {}U_1,,U_n $ with the same set of generators as \(S_\varepsilon^{-1}\mathcal{C\!F\!L}(L,\boldsymbol{s})\), and set \[C_\varepsilon:= \prod_{\boldsymbol{s}\in{\mathbb{H}}(L)} C_\varepsilon(\boldsymbol{s}).\] Each complex \(C_\varepsilon\) is equipped with a homological grading coming from \(\mathcal{C\!F\!L}(L)\)¹⁰.

Denote by \(\Lambda_j\) the \(j\)-th column vector of the surgery matrix \(\Lambda\). Under the identification \(H_1(S^3 \setminus L; {\mathbb{Z}}) = {\mathbb{Z}}^n\) induced by the meridians of the components of \(L\), the vector \(\Lambda_j\) represents the \(\lambda_j\)-framed longitude of \(K_j\). Given any arbitrarily (possibly empty) oriented sublink \(\vec{M}\) of \(L\), define \(\Lambda_{L,\vec{M}}\) to be the sum of the vectors \(\Lambda_j\) for which the orientation of \(K_j\) in \(\vec{M}\) is opposite to its orientation in \(L\). Each \(\varepsilon\in \mathbb{E}_n\) can be viewed as the indicator function of a sublink \(M \subset L\), where \(K_j \subset M\) if \(\varepsilon_j = 1.\) For each \(\vec{M}\) and \(\varepsilon,\) Manolescu-Ozsváth defined a map \[\label{eq:32phi95vecM} \phi^{\vec{M}}_{\varepsilon}: C_\varepsilon(\boldsymbol{s}) \rightarrow C_{\varepsilon'}(\boldsymbol{s}+ \Lambda_{L,\vec{M}} )\tag{6}\] where \(\varepsilon' - \varepsilon\) is the indicator function of \(M\).

It is proved in [34] that¹¹ for any fixed arbitrarily oriented sublink \(\vec{N} \subset L\), \[\sum_{\substack{\vec{M_1}\cup \vec{M_2}=\vec{N}\\\vec{M_1}\cap \vec{M_2}= \emptyset}} \phi^{\vec{M_2}}\circ \phi^{\vec{M_1}} = 0.\] When \(\vec{M}\) is the empty sublink, \(\phi^{\emptyset}_{\varepsilon}\) is the internal differential on \(C_\varepsilon(\boldsymbol{s})\). When \(|M|=1\), \(\phi^{\vec{M}}_{\varepsilon}\) is a chain map, and we denote by \(\Phi^{\vec{M}}_{\varepsilon}\) its induced map on the homology.

6.2 The \(H\)-function↩︎

By [34], when \(L\) is an \(L\)-space link we have that \(H_*(\mathcal{C\!F\!L}(L,\boldsymbol{s})) \cong {\mathbb{F}}[U]\) for each \(\boldsymbol{s}\in {\mathbb{H}}(L)\).

The \(H\)-function of \(L\) from [37] is defined by \[H_L(\boldsymbol{\mathbf{s}}) = -\tfrac{1}{2} gr(1),\] where \(1 \in {\mathbb{F}}[ U] \cong H_*(\mathcal{C\!F\!L}(L,\boldsymbol{s}))\).

The \(H\)-function of \(L\)-space links can be computed from the Alexander polynomials of all the sublinks by the work of Gorsky-Némethi. Recall that given a sublink \(L' \subset L\), Manolescu-Ozsváth defined the reduction map \(\pi_{L,L'}\colon {\mathbb{H}}(L)\to {\mathbb{H}}(L')\) to be \[s_i\mapsto s_i-\frac{\operatorname{lk}(K_i, L - L')}{2}\] for all \(i\) such that \(K_i\subset L'\).

In the above, \(\mathit{HFL}^-(L')\) denotes \(H_*(\mathcal{C\!F\!L}(L')/(Z_i: K_i\subset L'))\) and \(\boldsymbol{\mathbf{1}}\) denotes \((1,\dots, 1)\). Let \[\tilde{\Delta}_L(x_1,\dots,x_n) := \begin{cases} ( x_1\cdots x_n)^{\frac{1}{2}}\Delta_L(x_1,\dots,x_n) & \text{if n>1}\\ \Delta_L(x)/(1-x^{-1})& \text{if n=1}\\ \end{cases} \label{eq:normalized32Alex32poly}\tag{8}\] where \(\Delta_L(x_1,\dots,x_n)\) is the symmetrised Alexander polynomial. By [65], the following relation holds \[\tilde{\Delta}_{L}(x_1,\dots,x_n) = \sum_{\boldsymbol{\mathbf{s}}\in\mathbb{H}(L)}\chi(\mathit{HFL}^{-}(L,\boldsymbol{\mathbf{s}}))x_1^{s_1}\dots x_n^{s_n}, \label{eq:euler95char95Alex}\tag{9}\] where \(\boldsymbol{\mathbf{s}} = (s_1,\dots,s_n)\). For \(n=1\), we expand \(1/(1-x^{-1})\) as a power series in \(x^{-1}\). There is an overall sign ambiguity of \(\tilde{\Delta}_L\) which can be eliminated by the fact that \(H_L(\boldsymbol{\mathbf{s}})\) is non-increasing as \(\boldsymbol{\mathbf{s}}\) increases in each component.

Specialising to the case of a two-component \(L\)-space link \(L=K_1 \cup K_2\) where \(l=\operatorname{lk}(K_1,K_2)\), when \(s_1\) (resp. \(s_2\)) is sufficiently large, \(H_L(s_1,s_2)\) stabilises to a constant value denoted by \(H_L(\infty,s_2)\) (resp. \(H_L(s_1,\infty)\)). Moreover, there is a symmetry \[H_L(s_1,s_2)+s_1+s_2 = H_L(-s_1,-s_2).\] The following description of \(\Phi^{\vec{M}}_{\varepsilon}\) can be found in [25].

6.3 Zemke’s reformulation of the link surgery formula↩︎

In this section we review Zemke’s reformulation of the link surgery formula using bordered modules [35], which allows us to generalise the link surgery formula to rational coefficients.

We first recall Zemke’s definition of the type-\(D\) link surgery module from [35]. Given an oriented \(n\)-component link \(L \subset S^3\) and an integral framing \(\Lambda,\) Zemke constructed a type-\(D\) module \(\mathcal{X}_\Lambda(L)^{\mathcal{L}}\) which encodes all the maps \(\phi^{\vec{M}}_\varepsilon\) in the link surgery complex.

Suppose \(\{\boldsymbol{\mathbf{y}}_1,\dots,\boldsymbol{\mathbf{y}}_m\}\) is a free basis of \(\mathcal{C\!F\!L}(L)\). The underlying space of \(\mathcal{X}_\Lambda(L)^{\mathcal{L}}\) is \[\text{Span}_{ {\mathbb{F}}}\{\boldsymbol{\mathbf{y}}_1,\dots,\boldsymbol{\mathbf{y}}_m\} \otimes_{ {\mathbb{F}}} \boldsymbol{\mathbf{E}}_n.\] In other words, for each \(\varepsilon\in \mathbb{E}_n,\) and each free generator \(\boldsymbol{\mathbf{y}}\) of \(\mathcal{C\!F\!L}(L)\), we have a generator \(\boldsymbol{\mathbf{y}}^\varepsilon\).

We define the map \(\delta^1\) as follows. Fix \(\varepsilon\in \mathbb{E}_n,\) and let \(\boldsymbol{\mathbf{w}}^\varepsilon= \boldsymbol{\mathbf{y}}^\varepsilon_{i}\) for some \(i \in \{1,\dots,n\}\). Let \(N \subset L\) denote the sublink consisting of \(K_i\) for which \(\varepsilon_i =0.\) For each oriented sublink \(\vec{M}\subset N\) and \(\varepsilon'\in \mathbb{E}_n\) such that \(\varepsilon' - \varepsilon\) is the indicator function of \(M\), we can write \(\phi^{\vec{M}}_{\varepsilon}(\boldsymbol{\mathbf{w}}^\varepsilon) = \sum_{j=1}^m b_j \boldsymbol{\mathbf{y}}^{\varepsilon'}_j\), where we are considering \(\boldsymbol{\mathbf{w}}^\varepsilon\) as an element in \(C_\varepsilon\), and where \(b_j \in S^{-1}_{\varepsilon'} {\mathbb{F}}[Z_1,W_1,\dots,Z_n,W_n]=\boldsymbol{\mathbf{I}}_{\varepsilon'} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_{\varepsilon'}\).

Define an element \(t^{\vec{M}}\in \boldsymbol{\mathbf{I}}_{\varepsilon'} \cdot \mathcal{L}\cdot \boldsymbol{\mathbf{I}}_{\varepsilon}\) given by \(t^{\vec{M}} = \otimes_{i=1}^n t^{\vec{M}}_{i}\) where \[\label{eq:32def95t95M} t^{\vec{M}}_{i} := \begin{cases} 1 &\text{if K_i is not in M;}\\ \sigma_i &\text{if K_i is oriented the same way in \vec{M} as in L;}\\ \tau_i &\text{if K_i is oriented the opposite ways in \vec{M} as in L}. \end{cases}\tag{10}\]

We declare \(\delta^1(\boldsymbol{\mathbf{w}}^\varepsilon)\) to have the summand \[\sum_{j=1}^m \boldsymbol{\mathbf{y}}^{\varepsilon'}_j \otimes b_j \cdot t^{\vec{M}}.\]

In the case of integral framings, the box tensor pairing of bordered modules recovers the Manolescu-Ozsváth link surgery formula. In the following, \(\mathop{\mathrm{{\hat{\boxtimes}}}}\) denotes a completed version of the box tensor product defined in Section 5.2. See [35] for details.

Proposition 69, together with [34], implies that \(\boldsymbol{\mathbf{HF}}^{-}(S^3_{\Lambda}(L))\) can be recovered as the homology of \(\mathcal{X}_{\Lambda}(L)^\mathcal{L}\mathop{\mathrm{{\hat{\boxtimes}}}}{}_{\mathcal{L}}[\mathcal{D}_{(0,\dots,0)}]_{ {\mathbb{F}}[U_1,\dots, U_n]}\). In fact, by [44], as chain complexes over $ {}U,$ we have \[\mathcal{X}_{\Lambda}(L)^\mathcal{L}\mathop{\mathrm{{\hat{\boxtimes}}}}{}_{\mathcal{L}}[\mathcal{D}_{(0,\dots,0)}]_{ {\mathbb{F}}[U_1,\dots, U_n]} \simeq \boldsymbol{\mathbf{CF}}^{-}(S^3_{\Lambda}(L))\] where \(U\) acts by \(U_1\) on the left hand side.

It is pointed out in [35] that this can be generalised to allow for rational surgeries on links. We have the following, as a result of [34], [35], [44].

Since this statement does not appear in the literature, we include a proof at the end of this section for the reader’s convenience.

We begin by fixing the notation and setup. As observed in [68], a rational surgery on an oriented knot \(K \subset S^3\) can be viewed as a Morse surgery on a link in some rational homology sphere. Here, a Morse surgery means a Dehn surgery in which the meridian of each attached solid torus is identified with a longitude of the corresponding link component. We call the specified longitude the Morse framing. Explicitly, the \((\frac{p}{q})\)-surgery on \(K\subset S^3\) is realised by a surgery with coefficient \(d\) on the knot \(K \# O_{q/r} \subset L(q,r).\) Here \(d= \floor{\frac{p}{q}}\) and \(\frac{p}{q}= d + \frac{r}{q}\). The knot \(O_{q/r}\) is obtained from the Hopf link by performing \(-\frac{q}{r}\) on one component, viewing the other component as a knot in \(L(q,r) = S^3_{-q/r}(U)\). Denote by \(\lambda_d\) the Morse framing on \(O_{q/r}\) given by the \(d\) framed surgery. See Figure 42 for an example. Furthermore, we may associate to \((O_{q/r}, \lambda_d)\) a linear plumbing¹² \(\bar J := O \cup J \subset S^3\), whose oriented are oriented arbitrarily and with framing \(\Lambda=(d,a_1,\dots,a_m)\), where \[\frac{p}{q}=d-\cfrac{1}{a_1-\cfrac{1}{\ddots-\cfrac{1}{a_m}}},\] such that by performing surgeries on \(J\) according to the framing \[\Lambda_J=(a_1,\dots,a_m),\] the remaining component \(O\) becomes \(O_{q/r}\) with the desired Morse framing \(\lambda_d\).

In [44], the definition of the type-\(D\) module \(\mathcal{X}_\Lambda(L)^{\mathcal{L}}\) was extended to links in general \(3\)-manifolds, and more generally to bordered manifolds \(Y\) with torus is an invariant of \(Y\). As a consequence, we \(\mathcal{X}_\Lambda(L)^{\mathcal{L}}\) is natural with respect to performing Dehn surgeries, in the following sense:

Applying the above theorem to our setting, we have \[\label{eq:32linearplumbing95equivalence} \mathcal{X}_{\Lambda}(\bar J)^{\mathcal{L}_{m+1}} \boxtimes {}_{\mathcal{L}_m}\mathcal{D}_{(0,\dots,0)} \simeq \mathcal{X}_{\lambda_d}(O_{q/r})^\mathcal{K}\simeq \mathcal{D}_{p/q}^\mathcal{K}\tag{11}\] where \(\mathcal{D}_{p/q}^\mathcal{K}\) is the type-\(D\) module for rational framed solid torus defined in Section 5.4. Denote the type-\(D\) module on the left hand side of Equation 11 by \(\mathcal{Y}_{\Lambda}(\bar J)^\mathcal{K}\) and denote \[{}_{\mathcal{K}}\mathcal{Y}_{\Lambda}(\bar J) := \mathcal{Y}_{\Lambda}(\bar J)^\mathcal{K}\boxtimes {}_{\mathcal{K}|\mathcal{K}}[\mathbb{I}^{\Supset}].\]

Before proving Theorem 70 at the end of this section, we briefly discuss the behaviour of the structure maps under the chain homotopy equivalence in Equation 11 and prove a useful lemma. This relationship will play an important role in our study of the Spin\(^c\) decomposition of the rational link surgery complex in Section 6.4.

By the proof of [44], we have \[\begin{align} \tag{12} \mathcal{Y}_{\Lambda}(\bar J)^\mathcal{K}\boxtimes {}_{\mathcal{K}|\mathcal{K}}[\boldsymbol{\mathbf{I}}_0] &\simeq {}_{ {\mathbb{F}}[W,Z]}\mathcal{CFK}(O_{q/r}) \\ \tag{13} \mathcal{Y}_{\Lambda}(\bar J)^\mathcal{K}\boxtimes {}_{\mathcal{K}|\mathcal{K}}[\boldsymbol{\mathbf{I}}_1] &\simeq {}_{ {\mathbb{F}}[U,T,T^{-1}]}S_1^{-1}\mathcal{CFK}(O_{q/r}) \end{align}\] as type-\(A\) modules, where for \(\nu \in \{0,1\}\), the bimodule \({}_{\mathcal{K}|\mathcal{K}}[\boldsymbol{\mathbf{I}}_\nu]\) is obtained by restricting the identity bimodule \({}_{\mathcal{K}|\mathcal{K}}[\mathbb{I}^{\Supset}]\) to idempotent \(\boldsymbol{\mathbf{I}}_\nu\). We denote the complex in the left hand side of Equations 12 and 13 by \(\mathcal{Y}_0(\bar J)\) and \(\mathcal{Y}_1(\bar J)\), respectively.

For every \(\boldsymbol{t}\in\) Spin\(^c(L(q,r))\), \(\mathcal{CFK}(O_{q/r},\boldsymbol{t})\) is freely generated by a single generator with trivial differential. Furthermore, the Alexander grading on \(\mathcal{CFK}(O_{q/r})\) takes values in \(\frac{q-1}{2q}+\frac{1}{q} {\mathbb{Z}}\) and the Spin\(^c\) structure of a generator is determined by the Alexander grading via the map \[\begin{align} G_{L(q,r),O_{q/r}}: \;\frac{q}{2}+\frac{1}{q} {\mathbb{Z}}&\to{\mathbb{Z}}/q {\mathbb{Z}}\\ \frac{q-1}{2q}+\frac{k}{q} &\longmapsto k \end{align}\] defined in [68]¹³. Hence there is an isomorphism of the underlying space \[\boldsymbol{\mathbf{I}}_{\nu} \cdot {}_{\mathcal{K}}\mathcal{X}_{\lambda_d}(O_{q/r}) \cong \boldsymbol{\mathbf{I}}_{\nu} \cdot {}_{\mathcal{K}}\mathcal{D}_{p/q}\] sending the unique generator of \(S^{-1}_{\nu}\mathcal{CFK}(O_{q/r})\) in Alexander grading \(\frac{k}{q}\) to \(\boldsymbol{\mathbf{x}}_k^\nu\), and extending \(S^{-1}_{\nu} {\mathbb{F}}[W,Z]\)-equivariantly. We equip \({}_{\mathcal{K}}\mathcal{D}_{p/q}\) with the Alexander grading via this isomorphism. Let \[g: {}_{\mathcal{K}}\mathcal{Y}_{\Lambda}(\bar J) \to {}_{\mathcal{K}}\mathcal{D}_{p/q}\] denote the map obtained by postcomposing the chain homotopy equivalence in Equation 12 and 13 with the isomorphism of the underlying space above. We have the following. Recall the notation \(\bar J := O \cup J\).

We end this section by proving Theorem 70.

6.4 The rational link surgery formula↩︎

In this section we define the rational link surgery complex associated to a link \(L\subset S^3\) endowed with a rational framing, and we show with Theorem 75 that it can be used to compute the Heegaard Floer homology of the corresponding surgery on \(L\).

Let \(L=K_1\cup\cdots \cup K_n \subset S^3\) be an \(n\)-component link, and fix a set of surgery coefficients \(\Lambda = (\frac{p_1}{q_1},\dots,\frac{p_n}{q_n})\) where \(p_i \in {\mathbb{Z}}, q_i \in {\mathbb{Z}}_{>0}\) and \(p_i\) and \(q_i\) are coprime. Roughly speaking, the rational surgery complex is defined by assigning, for each lattice point \(\boldsymbol{s}\in {\mathbb{H}}(L)\) and each \(\varepsilon\in\mathbb{E}_n\), \(q_1\cdots q_n\) copies of the same complex \(C_\varepsilon(\boldsymbol{s})\), rather than a single copy as in the integral surgery complex.

We start by defining another lattice from $ {}(L)$ and a rational link surgery matrix.

Define a (rational) surgery matrix \(\tilde{\Lambda} = (\tilde{\Lambda}_{i,j})_{n\times n}\) by \[\begin{align} \tilde{\Lambda}_{i,j} = \begin{cases} p_i, &i=j \\ q_i\operatorname{lk}(K_i,K_j), \qquad & i\neq j \end{cases} \end{align}\] whose column vectors we denote \(\tilde{\Lambda}_j\) for \(j=1,\dots,n.\)

The complex \(C_{\Lambda}(L)\) splits over \(\tilde{ {\mathbb{H}}}(L)/H(\tilde{\Lambda})\), where \(H(\tilde{\Lambda})\) denotes the $ {}\(-span of the vectors\)_i.$ More precisely, if for each \(\boldsymbol{\mathbf{t}} \in \tilde{ {\mathbb{H}}}(L)/H(\tilde{\Lambda})\) we define \[C_{\Lambda}(L,\boldsymbol{\mathbf{t}}) := \bigoplus_{\varepsilon\in \mathbb{E}_n} \prod_{(\boldsymbol{s},\boldsymbol{k})\in \boldsymbol{\mathbf{t}}+H(\tilde{\Lambda})} C_\varepsilon(\boldsymbol{s},\boldsymbol{k}),\] then \(C_{\Lambda}(L,\boldsymbol{\mathbf{t}})\) is a summand of \(C_{\Lambda}(L)\).

Figure 43 shows an example of the simplified model of \(C_{\Lambda}(L)\), obtained by taking the homology of each \(C_{\varepsilon}(\boldsymbol{s},\boldsymbol{k})\) and considering only induced maps with \(|M|=1.\)

Observe that in the case of integral surgery coefficients, the above definitions recover the definitions of the integral link surgery complex.

We are now ready to state and prove the rational link surgery formula.

In the following, we prove the part of the statement of Theorem 75 regarding the splitting over the Spin\(^c\) structures.

6.5 A simplified model of \(C_{\Lambda}(L)\)↩︎

Following [21], we define a simplified model \(H_{\Lambda}(L)\) – referred to in [21] as the perturbed complex – of \(C_{\Lambda}(L)\) as follow. Recall from Definition 74 that when \(|M|=1,\) \[\tilde{\Phi}^{\vec{M}}_{\varepsilon}: H_*(C_\varepsilon(\boldsymbol{s},\boldsymbol{k})) \rightarrow H_*(C_{\varepsilon'}(\boldsymbol{s}',\boldsymbol{k}' ))\] are the maps induced by \(\tilde{\phi}^{\vec{M}}_{\varepsilon}\) on homology, where \(f(\boldsymbol{s}',\boldsymbol{k}') = f(\boldsymbol{s},\boldsymbol{k}) + \tilde{\Lambda}_{L,\vec{M}}\).

We now show \(C_{\Lambda}(L)\simeq H_{\Lambda}(L)\) for rational surgeries on two-component \(L\)-space links by applying Liu’s work [70].

Liu proved that one can replace the chain complex in each vertex of the hypercube by a chain homotopy equivalent one.

In particular, when \(n=2\), there is a simple description of the new hypercube.

We now prove the following theorem from the introduction. The argument is taken from [70].

7 Very good points and non \(L\)-space surgeries on \(L_{n,k}\)↩︎

In this section, we combine Gorsky-Némethi’s notion of “very good point” in [23], arguments in [25] and the rational link surgery formula to obtain Lemma 81, which provides an obstruction to \(L\)-space surgeries on two-component \(L\)-space links in terms of the \(H\)-function. Applying it to \(L_{n,k},\) we obtain the following result.

This result is used later in the proof of Theorem 3, where in order to classify \(L\)-space surgeries on \(L_{n,k}\), we need to obstruct certain rational surgeries from being \(L\)-space.

We first recall the rational link surgery formula using the simplified model described in Section 6.5, restricted to the case of a two-component \(L\)-space link \(L\). Fix any surgery coefficients \(\Lambda = (p_1/q_1,p_2/q_2)\) where \(p_i\in {\mathbb{Z}}, q_i\in {\mathbb{Z}}_{>0}\) and with \(p_i, q_i\) coprime for \(i=1,2\). Then, by Theorems 75 and 6, we have \(\widehat{\mathit{HF}} (S^3_{\Lambda}(L)) \cong H_*(H_{\Lambda}(L)/U)\). Here \(H_{\Lambda}(L)\) is the simplified surgery complex, with the underlying space given by \[H_{\Lambda}(L) = \bigoplus_{ \varepsilon\in \mathbb{E}_2} \prod_{(\boldsymbol{s},\boldsymbol{k})\in \tilde{ {\mathbb{H}}}(L)} H_*(C_{\varepsilon}(\boldsymbol{s},\boldsymbol{k})),\] where \(H_*(C_{\varepsilon}(\boldsymbol{s},\boldsymbol{k})) \cong {\mathbb{F}}\llbracket U\rrbracket\) for any \((\boldsymbol{s},\boldsymbol{k})\in \tilde{ {\mathbb{H}}}(L)\) and \(\varepsilon\in \mathbb{E}_2\) since \(L\) is an \(L\)-space link. Recall that the differential is the sum of \[\tilde{\Phi}^{\vec{M}}_{\varepsilon}: C_\varepsilon(\boldsymbol{s},\boldsymbol{k}) \rightarrow C_{\varepsilon'}(\boldsymbol{s}',\boldsymbol{k}' )\] over all oriented sublinks \(\vec{M}\subset L\) with \(|M|=1\), and where where \(f(\boldsymbol{s}',\boldsymbol{k}') = f(\boldsymbol{s},\boldsymbol{k}) + \tilde{\Lambda}_{L,\vec{M}}\) under the identification 19 . By definition, we have \(\tilde{\Phi}^{\vec{M}}_{\varepsilon} = \Phi^{\vec{M}}_{\varepsilon}\) as endomorphism of $ {}U$. All the non-trivial \(\Phi^{\vec{M}}_{\varepsilon}\) are described by Lemma 68.

7.1 Very good points↩︎

We recall Gorsky-Némethi’s definition of a very good point.

The following is our main technical lemma. The proof uses arguments similar to those in [25]; see, for example, the proof of [25].

7.2 The \(H\)-function of \(L_{n,k}\)↩︎

To apply Lemma 81, we need to compute the \(H\)-function of \(L_{n,k}.\) We recall from Proposition 24 that for \(n,k\geq 1\), the multivariable Alexander polynomial \(\Delta_{n,k}\) of \(L_{n,k}\) is \[\Delta_{n,k}\doteq\big{(}\sum_{i=0}^{k}x^{i}y^{k-i}\big{)}r_{n-1} -\big{(}\sum_{i=0}^{k-1}x^{i}y^{k-i-1}\big{)}r_n,\] where \(r_n=1+xy+\cdots+ x^ny^n\). The highest power for both \(x\) and \(y\) is \(n+k-1\) and the lowest power for both \(x\) and \(y\) is \(0\). Therefore the symmetrised Alexander polynomial is \[\Delta_{n,k}=(xy)^{-\frac{n+k-1}{2}}\Big(\big{(}\sum_{i=0}^{k}x^{i}y^{k-i}\big{)}r_{n-1} -\big{(}\sum_{i=0}^{k-1}x^{i}y^{k-i-1}\big{)}r_n\Big),\] up to a \(-\) sign. We may sketch \(\Delta_{n,k}\) in the \((i,j)\)-plane, where each term \(x^i y^j\) corresponds to a dot in the \((i,j)\)-coordinates, as depicted in Figure 44.

Indeed, the terms \[-(xy)^{-\frac{n+k-1}{2}}\left(\sum_{i=0}^{k-1}x^{i}y^{k-i-1}\right)\] have coordinates \((-\frac{n+k-1}{2},-\frac{n-k+1}{2}),\cdots,(-\frac{n-k+1}{2}, -\frac{n+k-1}{2})\), where from one term to the next the coordinates change by \((1,-1)\). They correspond to the \(k\) blue dots in the bottom left of the diagram. Multiplying by \(r_n\) changes the coordinates by \((i,i)\) for \(i=0,\dots,n\), which corresponds \(n+1\) copies of the diagonal sequence of \(k\) blue dots in total, each shifted by \((1,1)\) in coordinates.

Similarly, the terms \[(xy)^{-\frac{n+k-1}{2}}\left(\sum_{i=0}^{k}x^{i}y^{k-i}\right)\] have coordinates \((-\frac{n+k-1}{2},-\frac{n-k-1}{2}),\cdots,(-\frac{n-k-1}{2}, -\frac{n+k-1}{2})\) with coordinates change \((1,-1)\) between adjacent terms, which correspond to the \(k+1\) red dots in the bottom left of the diagram. Multiplying by \(r_{n-1}\) changes the coordinates by \((i,i)\) for \(i=0,\cdots,n-1\), corresponding to \(n\) copies of the diagonal sequence of \(k+1\) red dots, each shifted by \((1,1)\) in coordinates.

The full \(H\)-function \(H_{L_{n,k}}\) may be computed using the above description. However, for our application, we only need the following lemma. For simplicity, we suppress the index from \(H_{L_{n,k}}\).

8 Turaev torsion and \(L\)-spaces↩︎

In this section, we recall how the Turaev torsion can be used to determine the set of \(L\)-space surgery slopes on rational homology solid tori, following Rasmussen-Rasmussen and their main theorem in [45]. We start by recalling some properties of the Turaev torsion, and we refer the reader to [71] for a comprehensive discussion on the topic.

Let \(N\) be a compact orientable \(3\)-manifold whose boundary is a possibly empty union of tori, let \(H\) denote its first integer homology group and let \(Q(H)\) be the ring of fraction of $ {}[H]$, i.e. the localization of $ {}[H]$ by the multiplicative system of all non-zerodivisors.

The Turaev torsion (or maximal abelian torsion) \(\tau(N)\) of \(N\) is a topological invariant of \(N\) and it is an element of \(Q(H)\), well-defined up to multiplication by \(\pm H\), where \(H\) is identified with its image in \(Q(H)\) via the natural inclusion. We do not need the precise definition of this invariant, but only some of its properties.

First of all, it is possible to remove the ambiguity in the sign of the Turaev torsion by endowing \(N\) with a homological orientation, i.e. an orientation of the vector space \(H_{*}(N; {\mathbb{R}})\).

In some cases there is a natural choice of a homology orientation:

• if \(N\) is closed and oriented we define the natural homology orientation \(\omega_N\) of \(N\) as the one defined by a basis \((h_0, h_1, h_2, h_3)\) of \(H_*(N; {\mathbb{R}})\) such that \(h_i\) is a basis of \(H_i(N; {\mathbb{R}})\) and \(h_i\) and \(h_{3-i}\) are dual with respect to the non-degenerate pairing \(H_i(N; {\mathbb{R}})\times H_{3-i}(N ; {\mathbb{R}}) \rightarrow {\mathbb{R}}\) given by the orientation of \(N\). One can check that \(\omega_N\) does not depend on the choice of \((h_0, h_1, h_2, h_3)\);

• if \(N\) is the exterior of an oriented link \(L=K_1 \cup \cdots \cup K_m\) in a rational homology sphere \(M\), we define the natural homology orientation \(\omega_N\) of \(N\) to be the one determined by \(([\textit{pt}], [\mu_1],\dots, [\mu_m], [T_1], \dots, [T_{m-1}])\), where \([pt]\) is the class of a point in \(H_0(N; {\mathbb{R}})\) and \([\mu_i]\) and \([T_i]\) are the classes the oriented meridian of \(L_i\) and the oriented boundary of a tubular neighbourhood of \(K_i\) respectively. Here, as usual, the meridian is oriented so to have linking number \(1\) with \(K_i\) and \(T_i\) is oriented with the “outward vector first” convention. One can show that \(\omega_N\) does not depend on the numeration of the components of \(L\).

It is possible to eliminate also the ambiguity given by the multiplication by elements in \(H\), by fixing a so-called Euler structure on \(N\). We will not need do to this and we refer the reader to [71] for more details.

8.1 Turaev torsion and Alexander polynomial↩︎

We now suppose that the boundary of \(N\) is not empty. Denote by \(G=H/\text{Tors} H\) and let \(\text{pr}:Q(H)\rightarrow Q(G)\) denote the map induced by the projection \(H\rightarrow G\). Recall that the Alexander polynomial \(\Delta(M)\) is an element in $ {}[G]$, defined up to sign and multiplication by \(G\).

When \(b_1(N)=1\), \(N\) is a so-called a rational homology solid torus, since its boundary is a torus and it has the same rational homology groups of \(S^1\times D^2\). In this case we also fix an identification \(H= {\mathbb{Z}}\oplus \text{Tors}H\), and denote by \(t\) the element \((1,0)\), and by \(\tilde{t}\) its image in \(G\). The Turaev torsion and the Alexander polynomial are strongly related:

In particular, when \(H\) is torsion free and has rank at least \(2\) the Alexander polynomial and the Turaev torsion coincide. We will use this in the case of the exterior of links in \(S^3\).

In this case we notice that the Turaev torsion of \(N\) can be represented by an element in $ {}[H]$. This holds more generally:

Notice that when \(N\) is a rational homology solid torus the identification \(H= {\mathbb{Z}}\oplus \text{Tors}H\) defines a homology orientation of \(H\) obtained by choosing \(t\) as a generator of \(H_1(N; {\mathbb{R}})\). Therefore we can consider the sign-refined torsion associated to this homology orientation. It follows from the proof of [71] that, if we expand \((1-t)^{-1}\) as an infinite sum in positive powers of \(t\), then we can normalise \(\tau(N)\) as a formal sum \[\tau(N)=\sum_{\substack{h\in H \\ \text{pr}(h)\geq 0}}a_h h\] where \(a_h\) is an integer for all \(h\in H\), \(a_0\ne 0\) and \(a_h=1\) for all but finitely many \(h\in H\).

For example, when \(H\) has no torsion, it follows from Theorem 83 that the normalisation is \[\tau(N)=\frac{\Delta(N)}{1-t}\in{\mathbb{Z}}[[t]]\] where \((1-t)^{-1}\) is expanded as an infinite sum in positive powers of \(t\) and the Alexander polynomial \(\Delta(N)\) is normalised so that \(\Delta(N)\in {\mathbb{Z}}[t]\), \(\Delta(N)(0)\ne 0\) and \(\Delta(N)(1)=1\).

The last property of Turaev torsion we need is its behaviour under Dehn filling.

8.2 Turaev torsion and \(L\)-space filling slopes↩︎

The Turaev torsion will be useful for us because it contains all the information needed to determine the set of \(L\)-space surgeries on knots in rational homology spheres. We now briefly review how, following [45].

Let \(Y\) be a rational homology solid torus and let \(\mathit{Sl}(Y)\) denote the set of (rational) slopes in \(\partial Y\) \[\mathit{Sl}(Y)=\{ \alpha\in H_1(\partial Y; {\mathbb{Z}})\,| \,\alpha\text{ is primitive}\}/ \pm 1=\mathbb{P}(H_1(\partial Y; {\mathbb{Q}})).\] Each slope \([\alpha]\in \mathit{Sl}(Y)\) determines a Dehn filling of \(Y\), that we denote with \(Y(\alpha)\). Since \(Y\) is a rational homology solid torus, there exists a distinguished slope called homological longitude, that is defined as \([l]\), where \(l\in H_1(\partial Y; {\mathbb{Z}})\) is a generator of the kernel of the inclusion-induced map \[\iota:H_1(\partial Y; {\mathbb{Z}})\rightarrow H_1(Y; {\mathbb{Z}}).\] The element \(l\) is unique up to sign, and hence the class \([l]\in \mathit{Sl}(Y)\) is well-defined. Equivalently, \([l]\) can be defined as the unique slope defining a filling that is not a rational homology sphere. For example, when \(Y\) is the exterior of a knot \(K\) in \(S^3\), the homological longitude \([l]\) coincides with the slope defined by the (canonical) longitude of \(K\). We denote \(\mathit{Sl}^*(Y)=\mathit{Sl}(Y)\setminus [l]\).

We say that \([\alpha]\) is an \(L\)-space filling slope of \(Y\) if \(Y(\alpha)\) is an \(L\)-space, and we denote by \(L(Y)\) the set of \(L\)-space filling slopes of \(Y\). We say that \(Y\) is Floer simple if there exist at least two Dehn fillings of \(Y\) that are \(L\)-spaces, i.e. if \(|L(Y)|>1\).

It turns out that if \(Y\) is Floer simple, then it is possible to describe explicitly the set of \(L\)-spaces filling slopes of \(Y\) in terms of its Turaev torsion. Fix an identification \(H= {\mathbb{Z}}\oplus \text{Tors}H\) and write \[\tau(Y)=\sum_{\substack{h\in H_1(Y; {\mathbb{Z}}) \\ \text{pr}(h)\geq 0}}a_h h.\] We define the following subset of \(H_1(Y; {\mathbb{Z}})\) \[D^{\tau}_{>0}=\{x-y\,| x\notin S[\tau(Y)],\, y\in S[\tau(Y)] \text{ and } \text{pr}(x)>\text{pr}(y)\}\cap \iota(H_1(\partial Y;{\mathbb{Z}}))\] where \(S[\tau(Y)]=\{ h\in H_1(Y; {\mathbb{Z}})\,|\, a_h\ne 0 \}\) is the support of \(\tau(Y)\), and where \[\iota:H_1(\partial Y; {\mathbb{Z}})\rightarrow H_1(Y; {\mathbb{Z}})\] is induced by the inclusion \(\partial Y\hookrightarrow Y\).

We are now ready to state the main result of [45].

Recall that \([l]\in \mathit{Sl}(Y)\) denotes the homological longitude of \(Y\), and since the corresponding Dehn filling of \(Y\) is not a rational homology sphere, by the definition of \(L\)-space we have \([l]\notin L(Y)\). The second part of the statement of the theorem can be made more concrete as follows. Once we fix a basis \((\mu,\lambda)\) for \(H_1(\partial Y; {\mathbb{Z}})\), we can define a map \[\begin{array}{ccc} H_1(\partial Y; {\mathbb{Z}})&\longrightarrow&{\mathbb{Q}}\cup \{\infty\}\\ p\mu+q\lambda&\longmapsto&\frac{p}{q} \end{array}\] and this association induces an identification between \(\mathit{Sl}(Y)\) and $ {}{}\(. If the set\)D^{}_{>0}$ is not empty, we can apply this map to the set \(\iota^{-1}(D^{\tau}_{>0})\) and Theorem 86 states that if \(Y\) is Floer simple then \(L(Y)\) is a closed interval in \(\mathit{Sl}(Y)= {\mathbb{Q}}\cup \{\infty\}\) whose endpoints are consecutive elements in the image of \(\iota^{-1}(D^{\tau}_{>0})\) in $ {}{}$.

8.3 Multiple components and box-convexity↩︎

In this section, we observe that Theorem 86 implies a useful structural property of the set of \(L\)-space fillings slopes of a manifold with multiple boundary components.

Let \(N\) be the exterior of link in a rational homology sphere, with boundary components \(T_1,\dots, T_n\), and let \[\mathit{Sl}(N)=\mathbb{P}(H_1(T_1; {\mathbb{Q}}))\times \dots \times \mathbb{P}(H_1(T_n; {\mathbb{Q}}))\] be the set of (rational) multislopes in \(\partial N\). We denote \(L(N)\) the set of \(L\)-space fillings slopes of \(N\), and by \(\mathit{Sl}^*(N)\) the set of slopes that produce rational homology spheres by filling.

We will usually omit the prefix weakly; however “box-convex" should be understood to mean”weakly box-convex”. We define the box-convex hull of a subset \(S\subset \mathit{Sl}^*(N)\) as the smallest box-convex set containing \(S\).

We conclude the section with the following consequence of Proposition 89.

9 Classification of the \(L\)-space surgeries↩︎

In this section we classify the set of \(L\)-space surgeries on the links \(L_{n,k}\) and \(L'_{n,k}\). The classification for the links \(L_{n,k}\) is simpler, and is presented in Section 9.1. The majority of the section is devoted to the case of the links \(L'_{n,k}\), and is presented in Section 9.2.

We start by observing the following.

In particular, the previous lemma implies that the links \(L'_{n, n-1}\) are amphichiral.

9.1 Classification for the links \(L_{n,k}\)↩︎

We are now ready to prove Theorem 3, which we recall here.

We observe that, if \(n\ne 1\) and \(k\ne 1\), then there are surgeries on \(L_{n,k}\) that are non-\(L\)-space but are not covered by the result of Proposition 55. We can combine Theorem 3 with the main result of [33] to provide taut foliations on the remaining surgeries.

9.2 Classification for the links \(L'_{n,k}\)↩︎

In this section we prove Theorem  5, which classifies \(L\)-space surgeries on the (hyperbolic) links in the family \(L'_{n,k}\). We can reformulate the statement of Theorem 5 by using the language of box-convex sets introduced in the previous section. We denote \(l=n+k\).

We observe that the sets \(H_+/H_-\) lie on hyperbolas, while the sets \(L_+/L_-\) lie on lines.

The section is divided in two parts. In Section 9.2.1 we prove that \(L(L'_{n,k})\) contains the above mentioned box-convex hull, and then in Section 9.2.2 we show that it coincides with it.

We begin by identifying the hyperbolic links among the \(L'_{n,k}\). It is clear from the definition that \(L'_{n,k}\) is a torus link when \(k=0\), and by Lemma 91, this also happens when \(n=1\). On the other hand we have:

From now on, unless otherwise stated, we will suppose \(n\geq 2\) and \(k\geq 1\).

9.2.1 Finding \(L\)-space surgeries.↩︎

In this section we show that the set \(\mathcal{B}\cup H_+\cup H_-\cup L_+\cup L_-\), and hence its box-convex hull by Proposition 89, is contained in \(L(L'_{n,k})\). Since the components of \(L'_{n,k}\) are unknotted, we have \(\mathcal{B}\subset L(L'_{n,k})\), and the next proposition implies that the same holds for \(H_+\cup H_-\).

Our goal now is to show that also the set \(L_+\cup L_-\) is contained in \(L(L'_{n,k})\). This is the content of Proposition 98. We need some preliminary lemmas. We consider the link \(\mathcal{L}\) of Figure 50, and fix an integer number \(x\) satisfying \(0\leq x <n-1\). We denote by \(Y\) the rational homology solid torus \(S^3_{x,-1-x,\bullet}(\mathcal{L})\). As usual, we parametrise slopes on \(\partial Y\) by using the canonical meridian-longitude basis of the component \(K_3\) of \(\mathcal{L}\).

We conclude this section by finding more \(L\)-space surgeries on \(L'_{n,k}\).

9.2.2 Constraints on the set of \(L\)-space surgeries.↩︎

In this section we complete the proof of the classification Theorem 5. Recall that \(L'_{n,k}\) is hyperbolic if and only if \(n\geq 2\) and \(k\geq 1\), and this will be the assumption for the course of the section.

We denote \(L=L'_{n,k}\), and \(l=n+k\). Observe that, by virtue of Lemma 91, we can limit ourselves to study surgeries \(S^3_{r,s}(L)\) with \(r\leq 0\). Also notice that Theorem 5 holds when \(r=0\), since the manifold \(S^3_{0,s}(L)\) is an \(L\)-space for all \(s\ne 0\). For this reason, from now on we assume \(r<0\).

The proof is roughly divided in three parts. We write \(r=-\frac{p}{q}\) for coprime \(p,q>0\). We begin by assuming \(p\) coprime with \(l\), in which case the first homology group of \(S^3_{r,\bullet}(L)\) is isomorphic to \(\mathbb{Z}\). We prove Theorem 5 by computing the Turaev torsion of \(S^3_{r,\bullet}(L)\) and by using Theorem 86. We do this by studying the cases

• \(r<-(2n-1)\)

• \(r\in [-a,-a+1]\) with \(-a\in \mathbb{Z}\cap[1-2n,-2]\)

separately. Finally, we prove the theorem for general \(r<0\), by approximating \(r\) with rationals whose numerator is coprime with \(l\).

As premised, we fix \(r=\frac{p}{-q}\) with \(p,q\) coprime and \(p>0, q>0\), and suppose \(p\) coprime with \(l\). We fix canonical meridian-longitude bases \(\mu_i,\lambda_i\) for the components \(K_1\) and \(K_2\) of \(L\). In this case we have \[H_1(S^3_{r,\bullet}(L);{\mathbb{Z}})=\dfrac{<\mu_1, \mu_2>}{p\mu_1-ql\mu_2}\cong{\mathbb{Z}}\] and we fix the class of the element \(a\mu_1+ b\mu_2\) as positive generator of $ {}$, where \(a,b\;\in {\mathbb{Z}}\) satisfy \[\det\begin{pmatrix} p & a \\ -ql & b \end{pmatrix}=1.\]

The quotient map \(\pi:H_1(E_L; {\mathbb{Z}})\rightarrow H_1(S^3_{r,\bullet}(L); {\mathbb{Z}})\) in these coordinates is: \[\begin{array}{ccc}{\mathbb{Z}}^2&\overset{\pi}{\longrightarrow}& {\mathbb{Z}}\\ \mu_1=(1,0)&\longmapsto&ql\\ \mu_2=(0,1)&\longmapsto& p \end{array}\] and the induced map \(\pi_{\#}\) to the group rings is: \[\begin{array}{ccc}{\mathbb{Z}}[H_1(E_L; {\mathbb{Z}})]= {\mathbb{Z}}[x,y]&\overset{\pi_{\#}}{\longrightarrow}&{\mathbb{Z}}[H_1(S^3_{r,\bullet}(L);{\mathbb{Z}})]= {\mathbb{Z}}[t]\\ u(x,y)&\longmapsto&u(t^{ql}, t^p) \end{array}\]

We now assume \(r<-(2n-1)\). We recap the current assumptions and our goal.

We wanto to understand \(\iota^{-1}(D^{\tau}_{>0})\subset H_1(\partial S^3_{r,\bullet}(L); {\mathbb{Z}})\). We start by proving the following lemma.

We fix the canonical meridian and longitude basis \((\mu_2, \lambda_2)\) for the first homology group of the boundary of \(S^3_{r,\bullet}(L)\), and we use it to consider the image of \(\iota^{-1}(D^{\tau}_{>0})\) in $ {}{}$ via the map \(\varphi:x\mu_2+ y\lambda_2\mapsto \frac{x}{y}\).

The previous remark and Theorem 86 imply that in order to completely determine the set of \(L\)-space filling slopes of \(S^3_{r, \bullet}(L)\) it is enough to locate the maximum point of the image of \(\iota^{-1}(D^{\tau}_>0)\) in $ {}$. We denote this set by \(\mathcal{P}\). Thus our goal is to show that the element \[\rho=\dfrac{1}{\left\lfloor{\frac{r}{l}+\frac{2n-1}{l}}\right\rfloor}\] is the maximum element of \(\mathcal{P}\).

We first prove that \(\rho\) belongs \(\mathcal{P}\), and then that it is its maximum. Recall that we denote by \(\iota:H_1(\partial S^3_{r,\bullet}(L))\rightarrow H_1(S^3_{r,\bullet}(L))= {\mathbb{Z}}\) the map induced by the inclusion.

We now consider the class in \(H_1(\partial S^3_{r,\bullet}(L); {\mathbb{Z}})\) defined as \[u=-\mu_2 + \left\lceil{\frac{p}{ql}-\frac{2n-1}{l}}\right\rceil\lambda_2\]

We now focus our attention to the case when \(r\in [-a,-a+1)\) with \(-a\in {\mathbb{Z}}\cap [1-2n,-2]\). We summarise here our setting.

First of all, we show that we can assume \(a\in [1+n,2n-1]\).

From now on we assume \(a\in[1+n,2n-1]\). Analogously as we did before, we want to show that \(-1-2n+a\) is the maximum element of \(\mathcal{P}\). Recall that \(\mathcal{P}\) denotes the image of \(\iota^{-1}(D^{\tau}_>0)\) in $ {}$.

In order to deal with the case when \(p\) and \(l\) are not coprime, and conclude the proof in the general case, we need the following elementary lemma.

We can now complete the proof of Theorem 5.

Analogously to what happened in the case of the link \(L_{n,k}\), we have many non-\(L\)-space surgeries on \(L'_{n,k}\) that are not covered by the result Proposition 56. Even though we believe that the proof of Proposition 56 can be refined to produce a larger family of surgeries admitting taut foliations, we do not expect this approach to cover the entire set of non-\(L\)-space surgeries on \(L'_{n,k}\).

10 Applications to satellite knots↩︎

In this final section, we prove Theorem 12, which classifies \(L\)-space satellite knots whose pattern link is a two-bridge link.

Recall from the introduction that we denote by \(P(K,m)\) the \(m\)-twisted satellite knot of \(K\) with pattern \(P\). In this setting, the pattern link \(L_P=P \cup K_0\) is the two-component link in \(S^3\) obtained by viewing the solid torus containing \(P\) as a tubular neighbourhood of an unknot in \(S^3\), whose meridian we denote by \(K_0\). Since the case when \(L_P\) is a torus two-bridge link follows from the results of [16], [43], we focus on the hyperbolic case.

We start by fixing some notation. We denote by \(\mu_{K},\lambda_{K}\) the meridian and longitude of \(K\), and by \(\mu_0, \lambda_{0}\) and \(\mu_{P},\lambda_{P}\) the meridian and longitude of \(K_0\) and \(P\) respectively. We use these meridian-longitude bases to identify the slopes on the various boundary components with \(\overline{\mathbb{Q}}\). By construction, the exterior \(E_{P(K,m)}\) of \(P(K,m)\) is diffeomorphic to \(E_{L_P}\cup_{\varphi}E_{K}\), where the gluing map \[\varphi: \partial \nu K_0\rightarrow \partial \nu K\] satisfies \(\varphi(\mu_0)=m\mu_K+\lambda_K\) and \(\varphi(\lambda_0)=\mu_K\).

References↩︎

---

1. See, for instance, [27], [28].↩︎

2. The only two-bridge knots with non-meridional \(L\)-space surgeries are the \(T(2,k)\) torus knots.↩︎

3. The definition is a straightforward generalisation of that of persistently foliar knot, given by Delman and Roberts in [29]; see Definition 26.↩︎

4. We warn the reader the difference in notation with [32]: the link \(L(a_1,a_2,\dots,a_n)\) here corresponds to \(L(a_1,-a_2,\dots,(-1)^{n-1}a_n)\) there.↩︎

5. A similar discussion holds more generally for embedded plumbing graphs.↩︎

6. The proof presented there is for knots, but the same proof works also for links.↩︎

7. Notice that if \(|w(D_i^j)| = 2\) this is equivalent to asking that \(S\) intersects \(c_i^j\).↩︎

8. That is to say, it is a tree with one vertex of valency \(v\) and all the others of valency one.↩︎

9. We use this difference in notation since \(T\) is invertible in \(\boldsymbol{\mathbf{I}}_1 \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_1\) and in \(\boldsymbol{\mathbf{I}}_1 \cdot \mathcal{K}\cdot \boldsymbol{\mathbf{I}}_0\).↩︎

10. Observe that, in contrast to the usual case, \(C_\varepsilon\) is not the direct sum of its graded part, but we still refer to it as a graded complex.↩︎

11. Note the difference in notation: in [34] \(\Phi^{\vec{M}}\) is used to denote the map on the chain level.↩︎

12. i.e. a linear chain of unknots pairwise linked as Hopf links.↩︎

13. The Alexander grading determines a relative Spin\(^c\) structure \(\underline{\operatorname{Spin}}^c(L(q,r),O_{q/r})\), which determines a Spin\(^c\) structure of \(L(q,r)\) by [68]. For a computation of the Alexander gradings of \(O_{q/r}\) when \(r=1\), see [69]. A formula for arbitrary \(r\) follows from applying [69] recursively.↩︎

14. Observe that \(\boldsymbol{s}'\) agrees with \(\boldsymbol{s}+ \Lambda_{L,\vec{M}}\) in every \(i\)-th component for which \(\varepsilon_i=0\). It follows that \(C_{\varepsilon'}(\boldsymbol{s}') \cong C_{\varepsilon'}(\boldsymbol{s}+ \Lambda_{L,\vec{M}} )\).↩︎

15. We view \(H_*(C_\varepsilon)\) as a chain complex with vanishing differential and identify \(U\) with \(U_1\).↩︎

16. Here with a slight abuse of notations, we are identifying integers in \(\{0,\dots, q_i-1\}\) with their classes in $ {}/q_1 {}$.↩︎

17. Here we are using in the notation of Lisca–Stipsicz [31], where the Seifert fibered space \(M(e_0; r_1, r_2,\dots,r_k)\) is obtained by \(e_0\)–surgery on an unknot with \(k\) meridians having \(-\frac{1}{r_i}\) surgery coefficient on the \(i\)-th one.↩︎

18. Note that in general the longitude \(\lambda_P\) that we use to identify slopes on the boundary with \(\overline{\mathbb{Q}}\) does not coincide with the canonical longitude of the knot \(P(K,m)\). Nonetheless, the property of having large surgeries that are \(L\)-spaces does not depend on the choice of the longitude.↩︎

19. We point out that the statement of [76], which implies that \(K\) is an \(L\)-space knot cannot be directly applied here, since the authors in [76] use a slightly different definition of \(L\)-space knot (they do not require the \(L\)-space surgery to be positive).↩︎