1 Introduction↩︎

In [1], Morrison–Walker–Wedrich introduced the (Khovanov) skein lasagna module, a \(4\)-manifold invariant constructed using a version of \(\mathfrak{gl}_{2}\) Khovanov-Rozansky homology for links in \(S^{3}\). More precisely, given a compact oriented \(4\)-manifold \(X\) and a (possibly empty) framed oriented link \(L\subset\partial X\), the authors of [1] associate to the pair \((X,L)\) a triply-graded \(\mathbb{Q}\)-module denoted by \(\mathcal{S}_{0}^{2}(X;L)\). In recent years this invariant and its various extensions have been shown to detect exotic phenomena in dimension \(4\) in a purely algebraic manner [2]–[4].

The invariant also enjoys robust gluing properties [5]–[8] for facilitating computations. In particular, Manolescu–Walker–Wedrich [6] provide formulas for computing the Khovanov skein lasagna module of a 4-manifold after attaching an index \(k\) handle, \(k\in\{1,\dots,4\}\). For \(k=2\), the handle attaching formula expresses the invariant as a colimit of lasagna modules governed by a cabling pattern of the attaching link of the \(2\)-handle. In the case where a \(2\)-handle is attached to a 4-ball \(B^{4}\) along a framed link \(K\), the handle attaching formula provides an isomorphism between the skein lasagna module of the resulting \(2\)-handlebody and an invariant called the cabled Khovanov homology of \(K\) [5] (see also [9]).

In principle, the handle attachment formulas in [6] allow for the computation of the skein lasagna module for 4-manifolds built out of \(1\)- and \(2\)-handles. However, these formulas are computationally complex even in the simplest cases. For example, at present it is unknown how to compute the terms in the colimit computing \(\mathcal{S}_{0}^{2}(B^{4}):=\mathcal{S}_{0}^{2}(B^{4};\emptyset)\) via the Kirby diagram of \(B^{4}\) consisting of a canceling 1- and 2-handle (Figure 21).

Motivated by the desire for an invariant more well-suited for \(4\)-manifolds built out of \(1\)- and \(2\)-handles, Ren, Wedrich, Willis, Zhang and the second author [10] defined a new invariant called the 1-dimensional inputs (Khovanov) skein lasagna module, an invariant of smooth 4-manifolds constructed in the same spirit as the original [1], but with Rozansky–Willis homology as the input link homology theory. Introduced by Rozansky [11] and further developed by Willis [12], Rozansky–Willis homology (denoted by \(\mathrm {Kh}_{RW}(L)\)) is an invariant for links in connected sums of \(S^{1}\times S^{2}\). In [10] the authors constructed a \(\mathfrak{gl}_{2}\)-refinement of Rozansky-Willis homology and showed it satisfied the functoriality properties necessary to construct a skein lasagna module. One can extract from their construction two link homology theories which we denote in this paper by \(\mathrm{KhR}_{2,O}^{-}(L)\) and \(\mathrm{KhR}_{2,T}^{-}(L)\); these are defined for (framed, oriented) null-homologous and two-divisible links \(L\subset\sqcup_{i=1}^{k}\#^{m_{i}}S^{1}\times S^{2}\), respectively. These give rise to 1-dimensional inputs Khovanov skein lasagna modules \(\overline{\mathcal{S}}_{0}^{2,O}(X;L)\) and \(\overline{\mathcal{S}}_{0}^{2,T}(X;L)\).

In this work, we present handle attachment formulas for \(k\)-handles for \(k\in\{1,\dots,4\}\) for the invariants \(\overline{\mathcal{S}}_{0}^{2,\diamond}(X;L)\), \(\diamond\in\{O,T\}\). The index \(k\) handle attaching formulas for \(\overline{\mathcal{S}}_{0}^{2,\diamond}(X;L)\) and the original skein lasagna module \(\mathcal{S}_{0}^{2}(X;L)\) are most noticeably distinct when \(k=2\):

Consider a pair \((X,L)\) where \(X\) is built from attaching \(2\)-handles along an \(m\)-component link \(K=K_{1}\cup\cdots\cup K_{m}\subset\partial\natural^{n}S^{1}\times B^{3}\). Analogous to the construction in [5], for \(\diamond\in\{O,T\}\) one can define versions of cabled Rozansky-Willis homology for a pair of links \(K,L\subset\sqcup_{i=1}^{k}\#^{m_{i}}S^{1}\times S^{2}\), denoted by \(\underline{\mathrm{KhR}}^{-}_{2,\diamond}(K,L)\). In view of Manolescu-Neithalath’s \(2\)-handlebody formula, one would expect an isomorphism between \(\overline{\mathcal{S}}_{0}^{2,\diamond}(X,L)\) and \(\underline{\mathrm{KhR}}^{-}_{2,\diamond}(K,L)\).

However, there are lasagna fillings in the 1-dimensional inputs setting which admit isotopies (called lasso moves) that change the number of core-parallel sheets intersecting 2-handles (see Figure 1). This gives rise to an additional relation (called the lasso relation) on \(\underline{\mathrm{KhR}}^{-}_{2,\diamond}(K,L)\) that one must quotient by to obtain the correct isomorphism type of \(\overline{\mathcal{S}}_{0}^{2,\diamond}(X,L)\).

We briefly describe the lasso relation. Let \(\widetilde{\mathbb{1}}({1,1})\subset S^{1}\times{S^{2}}\) be the 2-component standard belt link as in Figure 3, and let \(i\in\{1,\dots,m\}\). By taking the connected sum of the pairs \((\#^{n}S^{1}\times{S^{2}},K)\) and \((S^{1}\times{S^{2}},\widetilde{\mathbb{1}}({1,1}))\) along \(K_{i}\subset K\) and \(\widetilde{\mathbb{1}}({1,0})\subset\widetilde{\mathbb{1}}({1,1})\), we obtain an \((m+1)\)-component link \(K(i)\subset{\#^{n+1}S^{1}\times{S^{2}}}\) as in Figure 5. The lasso relation is then given by identifying the images of two \(2\)-handle cobordism maps \[\Phi_{\diamond},\Psi_{\diamond}:\bigoplus_{i=1}^{m}\underline{\mathrm{KhR}}^{-}_{2,\diamond}(K(i),L)\to\underline{\mathrm{KhR}}^{-}_{2,\diamond}(K,L)\] defined in Section 4.3.

We now state the main theorem of this work:

1.1 Calculations↩︎

In this work, we use items [item:1] and [item:2] to perform partial computations of \(\overline{\mathcal{S}}_{0}^{2,\diamond}\) for disk bundles over \(S^{2}\), and a partial vanishing result for \(\Sigma_{g}\times{D^{2}}\) where \(\Sigma_{g}\) is a genus \(g\) surface. We also recalculate \(\overline{\mathcal{S}}_{0}^{2,\diamond}(B^{4})\) using a Kirby diagram of \(B^{4}\) consisting of a canceling 1- and 2-handle pair.

Recall that in [5], the authors compute the ordinary skein lasagna module of \(S^{2}\times D^{2}\) and exhibit an isomorphism \[\mathcal{S}^{2}_{0}(S^{2}\times D^{2})\cong\mathbb{Q}[A_{0},A_{0}^{-1},A_{1}]\] for some formal variables \(A_{0}\) and \(A_{1}\). We show that \(\overline{\mathcal{S}}^{2,O}_{0}(S^{2}\times D^{2})\) and \(\overline{\mathcal{S}}^{2,T}_{0}(S^{2}\times D^{2})\) are proper, non-trivial quotients of \(\mathcal{S}^{2}_{0}(S^{2}\times D^{2})\):

Let \(D(p)\) denote the \(D^{2}\)-bundle over \(S^{2}\) with Euler number \(p\). As in the case of the ordinary skein lasagna modules, the invariants \(\overline{\mathcal{S}}^{2,O}_{0}(D(p))\) and \(\overline{\mathcal{S}}^{2,T}_{0}(D(p))\) can distinguish between the cases \(p>0\) and \(p<0\):

Finally, consider the 4-manifold \(\Sigma_{g}\times{D^{2}}\) where \(\Sigma_{g}\) denotes the orientable surface of genus \(g\).

1.2 Future Directions↩︎

The computational techniques developed and used in this work suggest the following questions and further directions.

Without evidence to the contrary (and in view of Proposition 2), we pose the following conjecture:

Note that the above property does not hold for the ordinary skein lasagna invariants, as for example \(\mathcal{S}_{0}^{2}(S^{2}\times D^{2},\alpha)\) is infinitely generated (as a \(\mathbb{Q}\)-vector space) in each skein grading \(\alpha\in\mathbb{Z}\).

In analogy with the \(g=0\) case, the authors conjecture a negative answer for positive-degree disk bundles and an affirmative answer for disk bundles of non-positive degree. It seems plausible that one could generalize the methods from [13] to compute these groups directly in homological degree zero, although this is beyond the scope of this paper.

Note that this conjecture slightly contrasts what one would expect in view of the adjunction inequality from Seiberg–Witten theory ([14], [15]). From this point view, an affirmative answer to the conjecture would indicate that the skein lasagna package could potentially detect distinct exotic phenomena from the Seiberg–Witten invariants.

Next, given a link \(L\subset\#^{n}S^{1}\times S^{2}\) and \(k\in\mathbb{N}\), let \(L(k)\subset S^{3}\) denote the link obtained from \(L\) by replacing each zero-surgery curve with \(k\) full-twists. It was shown in [12] that (the original version of) Rozansky-Willis homology can be approximated by the Khovanov homology of \(L(k)\): more precisely, there exists an isomorphism \(\mathrm {Kh}^{h,*}_{RW}(L)\cong\mathrm {Kh}^{h,*}(L(\mathbf{k}))\) for all homological degrees \(h\) larger than some constant \(c(L,\mathbf{k})\) depending on \(L\) and \(k\) such that \(c(L,k)\to-\infty\) as \(k\to\infty\).

Analogous statements hold for the \(\mathfrak{gl}_{2}\)-analogues \(\mathrm{KhR}_{2,O}^{-}\) and \(\mathrm{KhR}_{2,O}^{-}\) on the level of groups, but it does not imply that these approximation schema are sufficiently functorial for, e.g., facilitating computations of the relevant skein lasagna invariants. It would therefore be desirable to lift this full-twist approximation scheme to the \(\mathfrak{gl}_{2}\)-setting:

In [2], Ren–Willis used Lee’s deformation of Khovanov homology ([16], [17]) in tandem with Manolescu-Neithalath’s \(2\)-handle formula ([5]) in order to develop vanishing/non-vanishing criteria for the ordinary skein lasagna modules of \(2\)-handlebodies, allowing them to detect families of exotic pairs of such manifolds.

The handle formulas proven in this paper should port over directly to analogues of \(1\)-dimensional inputs skein lasagna modules defined with respect to link homology theories other than Khovanov homology. For example, Chen [19] defined an analogue of the \(0\)-dimensional inputs skein lasagna module using link Floer homology \(\widehat{HFL}\) for links in \(S^{3}\) as the input TQFT. However, the invariant \(\widehat{HFL}\) is also functorial for links in closed 3-manifolds including connected sums of \(S^{1}\times{S^{2}}\).

Alternatively, one could define a new invariant of links in connected sums of \(S^{1}\times S^{2}\) by constructing an analogue of the Rozansky projector in knot Floer homology via infinite full twists (see [20] for a related construction) and attempt to construct a 1-dimensional inputs skein lasagna modules from this putative homology theory. It would be interesting to see how this potential theory would compare to one constructed directly from \(\widehat{HFL}\).

1.3 Acknowledgments↩︎

The authors would like to thank Remy Bohm, Eugene Gorsky, Maggie Miller, Trevor Oliveira-Smith, and Melissa Zhang for helpful discussions, as well as Lisa Piccirillo and Qiuyu Ren for comments on an earlier draft. The first author was partially supported by the Simons collaboration “New structures in low-dimensional topology”, and the second author by NSF grant no. DMS-2302305 during the preparation of this article. The authors would also like to thank the 2025 Trisectors Workshop for facilitating the working group which helped inspire this project.

2 Background↩︎

2.1 Notation and Conventions↩︎

The conventions for Khovanov homology used in this paper follow [10], with only minor aesthetic differences in notation as recorded below. We refer to the link invariant defined by Rozansky and extended by Willis as Rozansky-Willis homology and refer the reader to [11], [12] for further details and background about this invariant.

Throughout, we work over \(\mathbb{Q}\), and we let “\(\otimes\)" denote vertical stacking of tangles.

• For a framed oriented link \(L\subset S^{3}\) we denote by \(\mathrm{KhR}_{2}(L)\) the \(\mathfrak{gl}_{2}\)-Khovanov-Rozansky homology of \(L\).

  • For the \(0\)-framed unknot \(U\), we denote by \(\mathbf{1},\mathbf{X}\in\mathrm{KhR}_{2}(U)\) the standard generators which lie in bi-degrees \((0,-1)\) and \((0,1)\), respectively.

  • It was shown in [1] that the invariant \(\mathrm{KhR}_{2}\) extends to a symmetric monoidal functor \[\mathrm{KhR}_{2}:\boldsymbol{Links}_{0}\to fd\boldsymbol{Vect}_{\mathbb{Q}}^{\mathbb{Z}\times\mathbb{Z}}\] where:

    • \(\boldsymbol{Links}_{0}\) is the category of framed oriented links in abstract disjoint unions of \(S^{3}\), with morphisms given by pairs \((W,\Sigma)\) where \(W\) is a 4-manifold diffeomorphic to a disjoint union of 4-balls, with a disjoint union of 4-balls removed from each of them, and \(\Sigma\subset W\) is a properly embedded framed oriented surface.

    • \(fd\boldsymbol{Vect}_{\mathbb{Q}}^{\mathbb{Z}\times\mathbb{Z}}\) denotes the category of bigraded finite-dimensional \(\mathbb{Q}\)-vector spaces and homogeneous maps between them.

  • Given \(L\subset S^{3}\), we have an isomorphism \[\mathrm{KhR}_{2}^{h,q}(L)\cong\mathrm {Kh}^{h,-q-w(L)}(m(L))\] where \(w(L)\) denotes the writhe of of a diagram of \(L\) in which the given framing of \(L\) agrees with the blackboard framing, and \(m(L)\) denotes the mirror of \(L\).

• For \(\diamond\in\{O,T\}\), we have symmetric monoidal functors \[\begin{align} &\mathrm{KhR}_{2,\diamond}^{+}:\boldsymbol{Links}_{1,\diamond}^{\mathop{\mathrm{op}}}\to f\boldsymbol{Vect}_{\mathbb{Q}}^{\mathbb{Z}\times\mathbb{Z}} & &\mathrm{KhR}_{2,\diamond}^{-}:\boldsymbol{Links}_{1,\diamond}\to f\boldsymbol{Vect}_{\mathbb{Q}}^{\mathbb{Z}\times\mathbb{Z}} \end{align}\] Here:

  • \(\boldsymbol{Links}_{1,T}\) is the category of (framed, oriented) links in abstract disjoint unions of connected sums of \(S^{1}\times S^{2}\) (denoted by \(\boldsymbol{Links}_{1}\) in [10]) with \(2\)-divisible homology class, and \(\boldsymbol{Links}_{1,O}\subset\boldsymbol{Links}_{1,T}\) denotes the full subcategory of null-homologous links. Cobordisms in these categories can be decomposed into elementary cobordisms; see [10] for a comprehensive list of such cobordisms.

  • \(f\boldsymbol{Vect}_{\mathbb{Q}}^{\mathbb{Z}\times\mathbb{Z}}\) denotes the category of bigraded \(\mathbb{Q}\)-vector spaces, finite-dimensional in every bi-degree, and homogeneous maps between them. These invariants are described below.

• The type \(O\) (or unrenormalized) Rozansky-Willis homology \(\mathrm{KhR}_{2,O}^{+}(L)\) (denoted by \(\mathrm{KhR}_{2}^{+}(L)\) in [10]) for a null-homologous link \(L\subset\#^{n}S^1\times S^2\), is related to the usual Rozansky-Willis homology \(\mathrm {Kh}_{RW}(L)\) of \(L\) as defined in [11], [12] via an isomorphism \[\mathrm{KhR}_{2,O}^{+,h,q}(L)\cong(\mathrm {Kh}_{RW}^{-h,q+w(L)}(L))^{\vee}\] where \((\cdot)^{\vee}\) denotes the dual. In this paper we will use the convention that if \(L\) is not null-homologous, then we formally set \(\mathrm{KhR}_{2,O}^{+}(L)=0\).

• The type \(T\) (or renormalized) Rozansky-Willis homology, which we denote by \(\mathrm{KhR}_{2,T}^{+}(L)\) (denoted by \(\widetilde{\mathrm{KhR}}_{2}^{+}(L)\) in [10]) for a 2-divisible link \(L\subset\#^{n}S^{1}\times S^{2}\), is defined by \[\label{eq:KhR954395T95from95O} \mathrm{KhR}_{2,T}^{+,h,q}(L):=\mathrm{KhR}_{2,O}^{+,h+\frac{1}{2}w(L),q-\frac{1}{2}w(L)}(L).\tag{1}\] In this paper we will use the convention that if \(L\) is not 2-divisible, then we formally set \(\mathrm{KhR}_{2,T}^{+}(L)=0\). Note that the \(h\)- and \(q\)- gradings both take values in half-integers.

• There are dual versions of \(\mathrm{KhR}_{2,O}^{+}(L)\) and \(\mathrm{KhR}_{2,T}^{+}(L)\) defined by \[\mathrm{KhR}_{2,\diamond}^{-,h,q}(L):=(\mathrm{KhR}_{2,\diamond}^{+,-h,-q}(m(L)))^{\vee},\qquad\diamond\in\{O,T\}.\] By tracing through the definitions and noting that \(w(m(L))=-w(L)\), we see that \[\label{eq:KhR95-95T95from95O} \mathrm{KhR}_{2,T}^{-,h,q}(L)=\mathrm{KhR}_{2,O}^{-,h+\frac{1}{2}w(L),q-\frac{1}{2}w(L)}(L).\tag{2}\] In [10] the invariants \(\mathrm{KhR}_{2,O}^{-}(L)\) and \(\mathrm{KhR}_{2,T}^{-}(L)\) are denoted by \(\mathrm{KhR}_{2}^{-}(L)\) and \(\widetilde{\mathrm{KhR}}_{2}^{-}(L)\), respectively.

• If \(L\) is a link in \(S^{3}\), we will sometimes simply write \(\mathrm{KhR}_{2,\diamond}(L)\) to denote \(\mathrm{KhR}_{2,\diamond}^{-}(S^{3},L)\cong\mathrm{KhR}_{2,\diamond}^{+}(S^{3},L)\), with the observation that \(\mathrm{KhR}_{2,O}(L)=\mathrm{KhR}_{2}(L)\).

• When changing orientations of components of a link \(L\), we obtain a degree shift in the type \(O\)-theory – if \((L,\mathfrak{o})\) denotes the link \(L\) with a possibly different orientation \(\mathfrak{o}\) than the given one, we have a non-canonical isomorphism \[\mathrm{KhR}_{2,O}^{\pm,h,q}(L,\mathfrak{o})\cong\mathrm{KhR}_{2,O}^{\pm,h+\frac{1}{2}(w(L)-w(L,\mathfrak{o})),q-\frac{1}{2}(w(L)-w(L,\mathfrak{o}))}(L).\] (See [2], Equation 10.) As a consequence, we have (non-canonical) isomorphisms \[\label{eq:KhR95T95invariant95under95orientations} \mathrm{KhR}_{2,T}^{\pm,h,q}(L,\mathfrak{o})\cong\mathrm{KhR}_{2,T}^{\pm,h,q}(L,\mathfrak{o}')\tag{3}\] for any choices of orientations \(\mathfrak{o}\),\(\mathfrak{o}'\) on the components of a link \(L\).

• For completeness, we note that \[\begin{align} \begin{aligned} \label{eq:conventions} &\mathrm{KhR}^{-,h,q}_{2,O}(L)=(\mathrm{KhR}_{2,O}^{+,-h,-q}(m(L)))^{\vee} \\ &\qquad\qquad\qquad\qquad\cong((\mathrm {Kh}_{RW}^{h,-q+w(L)}(m(L)))^{\vee})^{\vee} =\mathrm {Kh}_{RW}^{h,-q+w(L)}(m(L)). \end{aligned} \end{align}\tag{4}\]

The invariants \(\mathrm{KhR}_{2,\diamond}^{-}\) for \(\diamond\in\{O,T\}\) are governed by certain categorified idempotents called Rozansky projectors, which are denoted by \(P_{n,0}\). For these objects we follow the conventions found in [10], [22]. In [22], Zhang and the second author show that the original skein lasagna module of the pair \((S^{2}\times{D^{2}},L)\) for \(L\subset\partial(S^{2}\times D^{2})\) recovers \(\mathrm{KhR}_{2,O}^{+}(L)\) up to an extra tensoral factor. However, the invariants \(\overline{\mathcal{S}}_{0}^{2,O}\) and \(\overline{\mathcal{S}}_{0}^{2,T}\) (defined later in Section 3.1) take the minus versions \(\mathrm{KhR}_{2,O}^{-}\) and \(\mathrm{KhR}_{2,T}^{-}\) as their respective input TQFTs.

We recall the cobar construction of the Rozansky projector as discussed in [23]:

The Rozansky projector \(P_{n,0}\) enjoys the following properties.

3 1-dimensional inputs skein lasagna modules↩︎

3.1 Skein Lasagna Modules with 0- and 1- Dimensional Inputs↩︎

For the duration of this paper, given a compact oriented 4-manifold \(X\) and a framed oriented link \(L\subset\partial X\) we will refer to the pair \((X,L)\) as a 4-manifold/boundary link pair.

In [1] the authors construct an invariant \(\mathcal{S}_{0}^{2}(X;L)\) of 4-manifold/boundary link pairs \((X,L)\). These take the form of \(\mathbb{Q}\)-vector spaces generated by lasagna fillings, whose underlying skeins each have an associated set of input manifolds given by a collection of neighborhoods of a finite disjoint set of points in the interior of the 4-manifold. As the input manifolds are points, we refer to this construction as the skein lasagna module with 0-dimensional inputs. For more background on skein lasagna modules with 0-dimensional inputs, we refer the reader to [1], [5], [6].

We recall the definition of the skein lasagna module with 1-dimensional inputs¹ as constructed in [10].

If \(L\) is the empty link, then we use the notation \(\mathcal{S}_{0}^{2}(X):=\mathcal{S}_{0}^{2}(X;\emptyset)\) (resp. \(\overline{\mathcal{S}}_{0}^{2,\diamond}(X):=\overline{\mathcal{S}}_{0}^{2}(X;\emptyset)\)).

The invariants \(\mathcal{S}_{0}^{2}(X;L)\) and \(\overline{\mathcal{S}}_{0}^{2,O}(X;L)\) come equipped with a tri-grading \((h,q,\alpha)\in\mathbb{Z}\times\mathbb{Z}\times H_{2}^{L}(X;\mathbb{Z})\), whereas the tri-grading on \(\overline{\mathcal{S}}_{0}^{2,T}(X;L)\) takes values in \(\frac{1}{2}\mathbb{Z}\times\frac{1}{2}\mathbb{Z}\times H_{2}^{L}(X;\mathbb{Z}/2)\). Here, for \(R=\mathbb{Z}\) or \(\mathbb{Z}/2\), \(H_{2}^{L}(X;R):=\partial^{-1}([L])\in H_{2}(X,L;R)\) where \(\partial\) is the connecting map in the long exact sequence for the pair \((X,L)\). In all of these theories, the \(h\)- and \(q\)-gradings descend from the corresponding input homology theories, whereas the third grading (called the skein grading) for an element represented by a lasagna filling \(F\) corresponds to the homology class of the underlying skein surface \(\Sigma\) of \(F\).

3.2 Properties↩︎

We now discuss several properties of 1-dimensional inputs skein lasagna modules. First, note that there always exist canonical maps \[\begin{align} \label{eq:maps95between95skein95lasagnas} &f_{0,\diamond}:\mathcal{S}_{0}^{2}(X;L)\to\overline{\mathcal{S}}_{0}^{2,\diamond}(X;L),\;\diamond\in\{O,T\}, &f_{O,T}:\overline{\mathcal{S}}_{0}^{2,O}(X;L)\to\overline{\mathcal{S}}_{0}^{2,T}(X;L) \end{align}\tag{6}\] defined as follows. The map \(f_{0,O}\) associates \(0\)-dimensional input lasagna fillings of \((X,L)\) to their corresponding type \(O\) \(1\)-dimensional input lasagna fillings under the canonical isomorphism of functors \[\mathrm{KhR}_{2,O}^{-}|_{\boldsymbol{Links}_{0}}\cong\mathrm{KhR}_{2}:\boldsymbol{Links}_{0}\to f\boldsymbol{Vect}_{\mathbb{Q}^{\mathbb{Z}\times\mathbb{Z}}}.\] This map is well-defined, as the relations defining \(\mathcal{S}_{0}^{2}(X;L)\) are a subset of those used to define \(\overline{\mathcal{S}}_{0}^{2,O}(X;L)\).

Next, given the canonical inclusion of categories \(\boldsymbol{Links}_{1,O}\hookrightarrow\boldsymbol{Links}_{1,T}\), consider the natural transformation of functors \(\eta:\mathrm{KhR}_{2,O}^{-}\Rightarrow\mathrm{KhR}_{2,T}^{-}|_{\boldsymbol{Links}_{1,O}}\) induced by the grading shift \((\cdot)\mapsto(tq^{-1})^{w(L)/2}(\cdot)\). We then define \(f_{O,T}\) to be the map which sends each lasagna filling \(F=(\Sigma,\{B_{i}\}_{i\in{I}},\{L_{i}\}_{i\in{I}},\{v_{i}\}_{i\in{I}})\) to its corresponding grading-shifted lasagna filling \(F'=(\Sigma,\{B_{i}\}_{i\in{I}},\{L_{i}\}_{i\in{I}},\{\eta(v_{i})\}_{i\in{I}})\). Again, this map is well-defined as the relations defining \(\overline{\mathcal{S}}_{0}^{2,O}(X;L)\) are (modulo grading shifts) a subset of those used to define \(\overline{\mathcal{S}}_{0}^{2,T}(X;L)\). The map \(f_{0,T}\) is then defined to be the composite \(f_{O,T}\circ f_{0,O}\).

These maps decompose along tri-gradings, in the sense that they split as direct sums of maps \[\begin{align} &f_{0,O}^{(h,q,\alpha)}:\mathcal{S}_{0,h,q}^{2}(X;L;\alpha)\to\overline{\mathcal{S}}_{0,h,q}^{2,O}(X;L;\alpha), \\ &f_{0,T}^{(h,q,\alpha)}:\mathcal{S}_{0,h,q}^{2}(X;L;\alpha)\to\overline{\mathcal{S}}_{0,h+\frac{1}{2}\alpha^{2},q-\frac{1}{2}\alpha^{2}}^{2,T}(X;L;\overline{\alpha}), \\ &f_{O,T}^{(h,q,\alpha)}:\overline{\mathcal{S}}_{0,h,q}^{2,O}(X;L;\alpha)\to\overline{\mathcal{S}}_{0,h+\frac{1}{2}\alpha^{2},q-\frac{1}{2}\alpha^{2}}^{2,T}(X;L;\overline{\alpha}), \end{align}\] where \(\overline{\alpha}\) denotes the mod \(2\) reduction of \(\alpha\).

The invariant \(\overline{\mathcal{S}}_{0}^{2,\diamond}(X,L)\), like the \(0\)-dimensional inputs version, satisfies more general gluing properties. Let \((X,L)\) be a 4-manifold/boundary link pair, and let \(Y\subset X\) be a properly embedding separating 3-manifold such that \(\partial Y\) intersects \(L\subset\partial X\) in some set of points \(P\). Let \((X_{1},T_{1})\) and \((X_{2},T_{2})\) denote the pairs obtained by cutting \((X,L)\) along \(Y\), where \(T_{i}\subset\partial X_{i}\) are framed oriented tangles with endpoints \(P\). Given any framed oriented tangle \(T_{0}\subset Y\) with \(\partial T_{0}=P\), we have an induced map \[\overline{\mathcal{S}}_{0}^{2,\diamond}(X_{1};T_{1}\cup T_{0})\otimes \overline{\mathcal{S}}_{0}^{2,\diamond}(X_{2};-T_{0}\cup T_{2})\to\overline{\mathcal{S}}_{0}^{2,\diamond}(X;L)\] obtained by gluing together lasagna fillings on each side (see [6] and [2], Section 2.2). We have the following analogue of ([2], Proposition 2.5):

From Proposition 14 we obtain the following corollary:

It will also be helpful to consider the above gluing map when one of the inputs is fixed. In particular: suppose \((X,L)\) is a 4-manifold/boundary link pair with \(\partial X=Y\), let \(W\) be a 4-dimensional cobordism from \(Y\) to some other 3-manifold \(Y'\), and \(\Sigma\subset W\) a surface cobordism from \(L\subset Y\) to some link \(L'\subset Y'\). Then we have an induced map \[\begin{align} \overline{\mathcal{S}}_{0}^{2,\diamond}(W;\Sigma):\overline{\mathcal{S}}_{0}^{2,\diamond}(X;L)&\to\overline{\mathcal{S}}_{0}^{2,\diamond}(X\cup_{Y} W;L') \\ F\mapsto \mathsf{gl}(F\otimes\Sigma), \end{align}\] where \(\Sigma\) is treated as a lasagna filling of the 4-manifold/boundary link pair \((W,-L\sqcup L')\). One special case of this is where \((W,\Sigma)\) is the mapping cylinder of a diffeomorphism of pairs \(\varphi:(Y,L)\to(Y',L')\). In this case we will denote the induced map simply by \(\overline{\mathcal{S}}_{0}^{2,\diamond}(\varphi)\).

4 2-handles↩︎

4.1 Braid Group Actions↩︎

Before discussing the 2-handle attachment formula for \(\overline{\mathcal{S}}_{0}^{2,\diamond}\), we remark on braid group actions on Rozansky-Willis homology. Let \(K_{n}\) denote the \(n\)-cable of the framed link \(K\) in the boundary of a 4-manifold \(X\), where all parallel strands are oriented in the same direction, then there is a natural action of the braid group \(\mathfrak{B}_{n}\) on \(\mathcal{S}_{0}^{2}(X;K_{n})\) that permutes the parallel strands of \(K_{n}\). More specifically, for each generator \(\sigma_{i}\in{\mathfrak{B}_{n}}\), there is an associated endomorphism \(\beta_{i}\in\mathrm{End}({\mathcal{S}_{0}^{2}(X;K_{n})})\) defined by pulling back the action of \(\sigma_{i}\) on an \(n\)-punctured \(D^{2}\times{S^{1}}\) to a tubular neighborhood of \(K\subset{\partial{X}}\). When \(X=B^{4}\), we remark that this action factors through the symmetric group \(\mathfrak{S}_{n}\) by Grigsby–Licata–Wehrli [24].

4.2 Cabled 1-dimensional inputs skein lasagna↩︎

Let \(X\) be a compact oriented four-manifold with boundary \(\partial X=Y\), let \(K=K_{1}\cup\cdots\cup K_{m}\subset Y\) be a framed oriented link, and let \(X'\) be the 4-manifold with boundary obtained by attaching \(m\) 4-dimensional 2-handles to \(Y\) along \(K\). Furthermore, let \(L\subset Y\) be a (possibly empty) framed oriented link disjoint from \(K\) and let \(L'\subset Y':=\partial X'\) denote the image of \(L\) after performing the 2-handle attachments. In this section we will describe how to calculate the 1-dimensional inputs skein lasagna module of \((X',L')\) in terms of a certain cabled 1-dimensional input skein lasagna module associated to the triple \((X,K,L)\).

Given \(\mathbf{k}^{\pm}=(k_{1}^{\pm},\cdots,k_{m}^{\pm})\in\mathbb{N}^{m}\), let \(K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\subset Y\) denote the framed oriented link obtained from \(K\cup L\) by replacing each component \(K_{i}\) of \(K\) with \(k_{i}^{+}+k_{i}^{-}\) parallel copies of itself, such that \(k_{i}^{-}\) of the copies are given the orientation opposite to the orientation on \(K\). For \(\mathbf{k}^{+},\mathbf{k}^{-}\in\mathbb{N}^{m}\) and \(\diamond\in\{O,T\}\), there is a natural action of \[\mathfrak{B}_{\mathbf{k}^{\pm}}:=\prod_{i=1}^{m}(\mathfrak{B}_{k_{i}^{+}}\times\mathfrak{B}_{k_{i}^{-}})\] on the 1-dimensional inputs skein lasagna module \(\overline{\mathcal{S}}_{0}^{2,\diamond}\big(X;K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\big)\) as in ([6], Section 3). Analogous to the cobordism maps defined in ([5], Section 3.1), we may consider the homomorphisms \[\psi_{\diamond,i}^{[j]}:\overline{\mathcal{S}}_{0}^{2,\diamond}\big(X;K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\big)\to \overline{\mathcal{S}}_{0}^{2,\diamond}\big(X;K(\mathbf{k}^{+}+\mathbf{e}_{i},\mathbf{k}^{-}+\mathbf{e}_{i})\cup L\big)\] induced by (possibly dotted) annuli that cobound oppositely oriented parallel strands for \(i\in\{1,\dots,m\}\), \(j\in{\{0,1\}}\), where \(\psi_{\diamond,i}^{[0]}\) denotes the undotted annulus cobordism (resp. \(\psi_{\diamond,i}^{[1]}\) is induced by the dotted annulus cobordism) ². The map \(\psi_{\diamond,i}^{[j]}\) changes the bigrading by \((0,2j)\).

Let \(R=\mathbb{Z}\) or \(\mathbb{Z}/2\), and let \(H_{2}^{L}(X;R)\) denote the set \(\partial^{-1}([L])\subseteq{H_{2}(X,L;R)}\) where \(\partial\) is the connecting map in the relative homology long exact sequence. As in [6], we can identify \(H_{2}^{L'}(X';R)\) with a subgroup of \(R^{m}\oplus H_{2}(X,K\cup L;R)\), and subsequently write any element of \(H_{2}^{L'}(X';\mathbb{Z})\) (respectively, \(H_{2}^{L'}(X';\mathbb{Z}/2)\)) as an element of the form \((\boldsymbol{\alpha},\eta)\) (resp., \((\overline{\boldsymbol{\alpha}},\overline{\eta})\)) with \(\boldsymbol{\alpha}\in\mathbb{Z}^{m}\), \(\eta\in H_{2}(X,K\cup L;\mathbb{Z})\), (resp, \(\overline{\boldsymbol{\alpha}}\in(\mathbb{Z}/2)^{m}\), \(\overline{\eta}\in H_{2}(X,K\cup L;\mathbb{Z}/2)\)).

Given \(\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{m})\in\mathbb{Z}^{m}\), we let \(\boldsymbol{\alpha}^{\pm}=(\alpha_{1}^{\pm},\dots,\alpha_{m}^{\pm})\in\mathbb{N}^{m}\) be given by \(\alpha_{i}^{\pm}=\max\{0,\pm\alpha_{i}\}\), \(i=1,\dots,m\). Suppose \(\mathbf{k}^{+},\mathbf{k}^{-}\in\mathbb{N}^{m}\) are such that:

1. if \(R=\mathbb{Z}\), then \(\mathbf{k}^{+}-\mathbf{k}^{-}=\boldsymbol{\alpha}\), and

2. if \(R=\mathbb{Z}/2\), then \(\mathbf{k}^{+}-\mathbf{k}^{-}\equiv\boldsymbol{\alpha}\pmod{2}\).

Let \(F_{\boldsymbol{\alpha}^{\pm},\mathbf{r};R}\) denote the composition \[\begin{align} \label{eq:eta95map} \begin{aligned} &H_{2}^{K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L}(X;R)\hookrightarrow H_{2}(X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L;R) \\ &\qquad\qquad\qquad\qquad\qquad\qquad\to H_{2}(X,\nu(K)\cup L;R)\cong H_{2}(X,K\cup L;R). \end{aligned} \end{align}\tag{7}\] For any class \((\boldsymbol{\alpha},\eta)\in H_{2}^{L'}(W;\mathbb{Z})\) (respectively, \((\overline{\boldsymbol{\alpha}},\overline{\eta})\in H_{2}^{L'}(W';\mathbb{Z}/2)\)) and any \(\mathbf{k}^{\pm}\in\mathbb{N}^{m}\) satisfying (C1) (respectively, (C2)), there exists a unique element \(\eta_{\mathbf{k}^{\pm}}\in H_{2}^{K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L}(X;\mathbb{Z})\) (resp., \(\overline{\eta}_{\mathbf{k}^{\pm}}\in H_{2}^{K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L}(X;\mathbb{Z}/2)\)) which is sent to \(\eta\) (resp., \(\overline{\eta}\)) under \(F_{\mathbf{k}^{\pm};\mathbb{Z}}\) (resp., \(F_{\mathbf{k}^{\pm};\mathbb{Z}/2}\)).

Note that, by construction ([10], discussion before Remark 3.5), the invariant \(\overline{\mathcal{S}}_{0}^{2,\diamond}\) recovers Rozansky-Willis homology when \(X=\natural^{n}S^{1}\times{B^{3}}\). In particular there is a natural isomorphism \[\overline{\mathcal{S}}_{0}^{2,\diamond}(\textstyle{\natural^{n}}S^{1}\times B^{3};L)\cong\mathrm{KhR}_{2,\diamond}^{-}(L)\] for any link \(L\subset\partial(\natural^{n}S^{1}\times B^{3})=\#^{n}S^{1}\times S^{2}\). Furthermore, by Proposition 15 the braid group action on \(\mathrm{KhR}_{2,\diamond}^{-}(L)\) factors through the symmetric group. Given \(\mathbf{k}^{+},\mathbf{k}^{-}\in\mathbb{N}^{m}\), we write \[\mathfrak{S}_{\mathbf{k}^{\pm}}:=\prod_{i=1}^{m}(\mathfrak{S}_{k_{i}^{+}}\times\mathfrak{S}_{k_{i}^{-}})\] in analogy with the corresponding notation for the braid group action.

In terms of homology classes, in this case we have that \(\eta\) is uniquely determined by \(\boldsymbol{\alpha}\) (and similarly in the \(\diamond=T\) case), and so we drop it from the notation. (See [6], discussion before Section 3.2.)

Note that \(\underline{\mathrm{KhR}}^{-}_{2,\diamond,\boldsymbol{\alpha}}(K,L)\neq 0\) only if \[\begin{align} &\sum_{i=1}^{m}\alpha_{i}[K_{i}]+[L]=0\in H_{1}(\#^{n}(S^{1}\times S^{2});\mathbb{Z}) &\text{if }\diamond=O, \tag{8} \\ &\sum_{i=1}^{m}\overline{\alpha}_{i}[K_{i}]+[L]=0\in H_{1}(\#^{n}(S^{1}\times S^{2});\mathbb{Z}/2) &\text{if }\diamond=T. \tag{9} \end{align}\]

4.3 The lasso relation↩︎

Let \((X^{\prime},L^{\prime})\) denote the 4-manifold and boundary link pair obtained by attaching some number of \(4\)-dimensional 2-handles to \((X,L)\). In this section we spell out the additional relation that we must impose on \(\underline{\overline{\mathcal{S}}}_{0}^{2,\diamond}(X;K,L)\) in order to ultimately relate this object to the invariant \(\overline{\mathcal{S}}_{0}^{2,\diamond}(X';L')\).

We denote by \(\widetilde{\mathbb{1}}({k,l})\subset S^{1}\times S^{2}\) the 0-framed standard belt link in \(S^{1}\times S^{2}\) with \(k\) positively-oriented and \(\ell\) negatively-oriented strands (see Figure 3.) Let \(X\) and \(K\cup L\subset Y=\partial X\) be as in the previous section, and let \(i\in\{1,\dots, m\}\). We define a new link \(K(i)\subset Y\#S^{1}\times S^{2}\) as follows: first consider the link \[\widetilde{K}(i):=K\sqcup\widetilde{\mathbb{1}}({1,1})\subset Y\#S^{1}\times S^{2},\] where we have a disjoint copy of \(K\) inside of \(Y\setminus B^{3}\subset Y\#S^{1}\times S^{2}\), along with a disjoint copy of \(\widetilde{\mathbb{1}}({1,1})\subset S^{1}\times S^{2}\setminus B^{3}\subset Y\#S^{1}\times S^{2}\). Choose a framed arc \(a\) connecting the \(i\)-th component \(K_{i}\subset K\) with the positively oriented component \(\widetilde{\mathbb{1}}({1,0})\subset\widetilde{\mathbb{1}}({1,1})\). We then define \(K(i)\subset Y\# S^{1}\times S^{2}\) to be the framed link obtained by performing a (framed, oriented) connected sum along \(a\). More explicitly, we can write \(K(i)=K(i)_{1}\cup\cdots\cup K(i)_{m+1}\) where \[K(i)_{j}= \left\{ \begin{array}{ll} K_{i}\#\widetilde{\mathbb{1}}({1,0}) & ifj=i \\ \widetilde{\mathbb{1}}({0,1}) & ifj=m+1 \\ K_{j} & ifj\neq i,m+1. \end{array} \right.\] By isotoping \(a\) if necessary, we can assume that the connect sum region is disjoint from \(L\subset Y\setminus B^{3}\subset Y\# S^{1}\times S^{2}\), and hence \(K(i)\cup L\) is a well-defined link in \(Y\# S^{1}\times S^{2}\), independent of the choice of arc \(a\) up to diffeomorphism of pairs.

Next, we construct two cobordisms of pairs \[(W_{i},\Sigma_{i}),(W'_{i},\Sigma'_{i}):(Y\#(S^{1}\times S^{2}),K(i)\cup L)\to(Y,K\cup L)\] as follows: let \((Z_{i},S_{i})\) and \((Z'_{i},S'_{i})\) be the cobordisms of pairs given by attaching 4-dimensional 2-handles along parallel copies \(\gamma\) and \(\gamma'\) of \(K(i)_{m+1}=\widetilde{\mathbb{1}}({0,1})\) and \(K(i)_{i}=K_{i}\#\widetilde{\mathbb{1}}({1,0})\), respectively, with surfaces \(S_{i},S'_{i}\) given by the corresponding traces of \(K(i)\cup{L}\) in each cobordism. Since \(K(i)_{m+1}\) and \(K(i)_{i}\) each have geometric intersection number \(1\) with the \(S^{1}\times S^{2}\)-factor, it follows that \(Z_{i}\) and \(Z'_{i}\) are both cobordisms from \(Y\#S^{1}\times S^{2}\) to \(Y\).

We observe that \[S_{i}|_{Y}=S'_{i}|_{Y}=K\cup U\cup L\subset Y,\] where \(U=(U,0)\) denotes a 0-framed unknot contained in a \(3\)-ball. The surfaces \(S_{i}\) and \(S_{i}'\) each consist of a union of cylinders of the following forms:

1. Trivial cylinders connecting the components of \((K(i)\cup L)\setminus(K(i)_{m+1}\cup K(i)_{i})\) to their corresponding components in \((K\cup L)\setminus K_{i}\).

2. In \(S_{i}\): a pair of cylinders connecting \(K(i)_{m+1}\) to \(U\) and \(K(i)_{i}\) to \(K_{i}\).

3. In \(S_{i}'\): a pair of cylinders connecting \(K(i)_{i}\) to \(U\) and \(K(i)_{m+1}\) to \(K_{i}\).

One can see that the above description holds by noting that \(U\) can be obtained by sliding \(K(i)_{m+1}\) over the 2-handle attached along \(\gamma\), as well as by sliding \(K(i)_{i}\) over the 2-handle attached along \(\gamma'\) (see Figure 4).

Let \[\big(Y\times[0,1],C\big):\big(Y,K\cup U\cup L\big)\to\big(Y,K\cup L\big)\] denote the cobordism of pairs given by capping off the unknot component via a disk contained in \(B^{3}\times[0,\frac{1}{2}]\subset Y\times[0,1]\), and attaching trivial cylinders to remaining link components. We then define \[\begin{align} &(W_{i},\Sigma_{i}):=(Y\times[0,1],C)\circ(Z_{i},S_{i}), & &(W'_{i},\Sigma'_{i}):=(Y\times[0,1],C)\circ(Z'_{i},S'_{i}). \end{align}\] Now given \((\boldsymbol{\alpha},\eta)\in H_{2}^{L'}(X';\mathbb{Z})<\mathbb{Z}^{m}\oplus H_{2}(X,K\cup L;\mathbb{Z})\), one can show that there exists a unique element \(\eta_{i}\in H_{2}(X\natural S^{1}\times B^{3},K(i)\cup L;\mathbb{Z})\) such that \(\eta\) is sent to \((\eta_{i},[\Sigma_{i}])\) and \((\eta_{i},[\Sigma'_{i}])\) under the natural maps \[\begin{align} &H_{2}(X,K\cup L;\mathbb{Z})\to H_{2}(X\natural S^{1}\times B^{3},K(i)\cup L;\mathbb{Z})\oplus H_{2}(W_{i},-(K(i)\cup L)\sqcup (K\cup L);\mathbb{Z}), \\ &H_{2}(X,K\cup L;\mathbb{Z})\to H_{2}(X\natural S^{1}\times B^{3},K(i)\cup L;\mathbb{Z})\oplus H_{2}(W'_{i},-(K(i)\cup L)\sqcup (K\cup L);\mathbb{Z}), \end{align}\] respectively. In the mod \(2\) homology case, an element \((\overline{\boldsymbol{\alpha}},\overline{\eta})\in H_{2}^{L'}(X';\mathbb{Z}/2)\) gives rise to an analogously characterized element \(\overline{\eta}_{i}\in H_{2}(X\natural S^{1}\times B^{3},K(i)\cup L;\mathbb{Z}/2)\).

Next, we extend the cobordism maps and \(K(i)\)’s to cables. Let \(\mathbf{k}^{+},\mathbf{k}^{-}\in\mathbb{N}^{m}\), \(\ell^{+},\ell^{-}\in\mathbb{N}\), and consider the \((m+1)\)-tuples \((\mathbf{k}^{\pm},\ell^{\pm}):=(k_{1}^{\pm},\dots,k_{m}^{\pm},\ell^{\pm})\in\mathbb{N}^{m+1}\). Given \(i\in\{1,\dots, m\}\), we define \[K(\mathbf{k}^{\pm};\ell^{\pm};i):=K(i)((\mathbf{k}^{+},\ell^{+}),(\mathbf{k}^{-},\ell^{-}))\subset Y\#S^{1}\times S^{2}.\] (See Figure 5.)

We have analogously defined corbordisms of pairs \[\begin{align} &(W_{i},\Sigma_{\mathbf{k}^{\pm},\ell^{\pm},i}):\big(Y\#S^{1}\times S^{2},K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L\big) \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\to\big(Y,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\big), \\ &(W'_{i},\Sigma'_{\mathbf{k}^{\pm},\ell^{\pm},i}):\big(Y\#S^{1}\times S^{2},K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L\big) \\ &\qquad\qquad\qquad\qquad\to\big(Y,K(\mathbf{k}^{+}+(\ell^{+}-k_{i}^{+})\mathbf{e}_{i},\mathbf{k}^{-}+(\ell^{-}-k_{i}^{-})\mathbf{e}_{i})\cup L\big), \end{align}\] obtained by appropriately taking framed push-offs of the surface components of the cobordisms \((W_{i},\Sigma_{i})\) and \((W'_{i},\Sigma'_{i})\) defined above, with analogous decompositions \[\begin{align} &(W_{i},\Sigma_{\mathbf{k}^{\pm},\ell^{\pm},i})=(Y\times[0,1],C_{\ell^{\pm}})\circ(Z_{i},S_{\mathbf{k}^{\pm},\ell^{\pm},i}), \tag{10} \\ &(W'_{i},\Sigma'_{\mathbf{k}^{\pm},\ell^{\pm},i})=(Y\times[0,1],C_{k_{i}^{\pm}})\circ(Z'_{i},S'_{\mathbf{k}^{\pm},\ell^{\pm},i}). \tag{11} \end{align}\] Now let \(\diamond\in\{O,T\}\). For the rest of this section we will denote \[\begin{align} &\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm}):=\overline{\mathcal{S}}_{0}^{2,\diamond}\big(X;K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\big)\big\{-|\mathbf{k}^{+}|-|\mathbf{k}^{-}|\big\}, \\ &\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\ell^{\pm};i):=\overline{\mathcal{S}}_{0}^{2,\diamond}\big(X\natural S^{1}\times B^{3};K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L\big)\big\{-|\mathbf{k}^{+}|-|\mathbf{k}^{-}|-\ell^{+}-\ell^{-}\big\}. \end{align}\] Next, consider the maps \[\begin{align} &\Phi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}:=\overline{\mathcal{S}}_{0}^{2,\diamond}(W_{i},\Sigma_{\mathbf{k}^{\pm},\ell^{\pm},i}):\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\ell^{\pm};i)\to\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm}) \tag{12} \\ &\Psi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}:=\overline{\mathcal{S}}_{0}^{2,\diamond}(W'_{i},\Sigma'_{\mathbf{k}^{\pm},\ell^{\pm},i}):\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\ell^{\pm};i)\to\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm}+(\ell^{\pm}-k_{i}^{\pm})\mathbf{e}_{i}) \tag{13} \end{align}\] obtained as the gluing maps associated to \((W_{i},\Sigma_{\mathbf{k}^{\pm},\ell^{\pm},i})\) and \((W'_{i},\Sigma'_{\mathbf{k}^{\pm},\ell^{\pm},i})\), respectively. (See Section 3.2).

Let \(\eta_{\mathbf{k}^{\pm},\ell^{\pm},i}\in H_{2}^{K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L}(X\natural S^{1}\times B^{3};\mathbb{Z})\) be the unique element which is sent to \(\eta_{i}\in H_{2}(X\natural (S^{1}\times B^{3}),K(i)\cup L;\mathbb{Z})\) under the analogue of the map from (7 ). Then one can show that \(\eta_{\mathbf{k}^{\pm}}\) and \(\eta_{\mathbf{k}^{\pm}+(\ell^{\pm}-k_{i}^{\pm})\mathbf{e}_{i}}\) are sent to \((\eta_{\mathbf{k}^{\pm},\ell^{\pm},i},[\Sigma_{\mathbf{k}^{\pm},\ell^{\pm},i}])\) and \((\eta_{\mathbf{k}^{\pm},\ell^{\pm},i},[\Sigma'_{\mathbf{k}^{\pm},\ell^{\pm},i}])\) under the natural maps \[\begin{align} &H_{2}^{K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L}(X;\mathbb{Z})\to \\ &\qquad\qquad H_{2}(X\natural S^{1}\times B^{3},K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L;\mathbb{Z}) \\ &\qquad\qquad\qquad\oplus H_{2}(W,-(K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L)\sqcup (K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L);\mathbb{Z}), \\ &H_{2}^{K(\mathbf{k}^{+}+(\ell^{+}-k_{i}^{+})\mathbf{e}_{i},\mathbf{k}^{-}+(\ell^{-}-k_{i}^{-})\mathbf{e}_{i})\cup L}(X;\mathbb{Z})\to \\ &\qquad\qquad H_{2}(X\natural S^{1}\times B^{3},K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L;\mathbb{Z}) \\ &\qquad\qquad\qquad\oplus H_{2}(W',-(K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L)\sqcup (K(\mathbf{k}^{+}+(\ell^{+}-k_{i}^{+})\mathbf{e}_{i},\mathbf{k}^{-}+(\ell^{-}-k_{i}^{-})\mathbf{e}_{i})\cup L);\mathbb{Z}), \end{align}\] respectively, with a similar statement in the mod \(2\) homology case. In particular, we see that \(\Phi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}\) and \(\Psi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}\) decompose along skein gradings as follows: \[\begin{align} &\Phi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}^{\eta}:\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\ell^{\pm};i;\eta_{\mathbf{k}^{\pm},\ell^{\pm},i})\to\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\eta_{\mathbf{k}^{\pm}}) \\ &\Psi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}^{\eta}:\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\ell^{\pm};i;\eta_{\mathbf{k}^{\pm},\ell^{\pm},i})\to\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm}+(\ell^{\pm}-k_{i}^{\pm})\mathbf{e}_{i};\eta_{\mathbf{k}^{\pm}+(\ell^{\pm}-k_{i}^{\pm})\mathbf{e}_{i}}). \end{align}\]

4.4 A 2-handle formula↩︎

In this section, we construct an isomorphism between the 1-dimensional inputs skein lasagna module of a 4-manifold with a 2-handle attached and a quotient of the cabled 1-dimensional inputs skein lasagna module.

Consider the action of the braid group \[\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm},i}:=\mathfrak{B}_{(\mathbf{k}^{\pm},\ell^{\pm})}\cong\mathfrak{B}_{\mathbf{k}^{\pm}}\times\mathfrak{B}_{\ell^{+}}\times\mathfrak{B}_{\ell^{-}}\] on \(K(\mathbf{k}^{\pm};\ell^{\pm};i)\). We have natural projection maps \[\begin{align} &p_{\mathbf{k}^{\pm},\ell^{\pm},i}:\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm},i}\to\mathfrak{B}_{\mathbf{k}^{\pm}}, & &p'_{\mathbf{k}^{\pm},\ell^{\pm},i}:\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm},i}\to\mathfrak{B}_{\mathbf{k}^{\pm}+(\ell^{\pm}-k_{i}^{\pm})\mathbf{e}_{i}}, \end{align}\] where \(p_{\mathbf{k}^{\pm},\ell^{\pm},i}\) is given by the projection onto the \(\mathfrak{B}_{\mathbf{k}^{\pm}}\) factor, and \(p'_{\mathbf{k}^{\pm},\ell^{\pm},i}\) is projection onto the factor \[(\prod_{\substack{j=1 \\ j\neq i}}^{m}\mathfrak{B}_{k_{j}^{+}})\times(\prod_{\substack{j=1 \\ j\neq i}}^{m}\mathfrak{B}_{k_{j}^{-}})\times\mathfrak{B}_{\ell^{+}}\times\mathfrak{B}_{\ell^{-}}\cong\mathfrak{B}_{\mathbf{k}^{\pm}+(\ell^{\pm}-k_{i}^{\pm})\mathbf{e}_{i}}.\] Furthermore, for \(j=1,\dots,m\), and \(k=0,1\), the annulus homomorphisms take the form of maps \[\psi_{\diamond,j}^{[k]}:\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\ell^{\pm};i)\to\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm}+\mathbf{e}_{j};\ell^{\pm};i),\] whereas for \(j=m+1\), the annulus homomorphisms for \(k=0,1\) are of the form \[\psi_{\diamond,m+1}^{[k]}:\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\ell^{\pm};i)\to\overline{\mathcal{S}}_{0}^{2,\diamond}(\mathbf{k}^{\pm};\ell^{\pm}+1;i).\] Each of these maps are \(\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm},i}\)-equivariant, where the actions on the codomains of \(\psi_{\diamond,j}^{[k]}\), \(j=1,\dots,m\) and \(\psi_{\diamond,i,m+1}^{[k]}\) respectively, \(k=0,1\), are induced by the canonical inclusions \[\begin{align} &i_{\mathbf{k}^{\pm},\ell^{\pm},i;j}:\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm},i}\hookrightarrow\mathfrak{B}_{\mathbf{k}^{\pm}+\mathbf{e}_{j},\ell^{\pm},i}, & &i_{\mathbf{k}^{\pm},\ell^{\pm},i;m+1}:\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm},i}\hookrightarrow\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm}+1,i}. \end{align}\] By construction we see that \(\Phi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}\) and \(\Psi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}\) are both \(\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm},i}\)-equivariant, where the action of \(\mathfrak{B}_{\mathbf{k}^{\pm},\ell^{\pm},i}\) on their respective codomains are induced by the projections \(p_{\mathbf{k}^{\pm},\ell^{\pm},i}\) and \(p'_{\mathbf{k}^{\pm},\ell^{\pm},i}\). Furthermore, these maps fit into the following commutative diagrams for \(i=1,\dots,m\), \(j=1,\dots,m\), \(j\neq i\), \(k=0,1\) (and decompose along skein gradings appropriately):

In particular, this implies that the collections of maps \(\{\Phi^{\eta}_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}\}\) and \(\{\Psi^{\eta}_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}\}\) are compatible with the relations defining the cabled 1-dimensional inputs skein lasagna modules, and so induce well-defined maps \[\begin{align} &\Phi^{(\boldsymbol{\alpha},\eta)}_{O},\Psi^{(\boldsymbol{\alpha},\eta)}_{O}:\bigoplus_{i=1}^{m}\underline{\overline{\mathcal{S}}}^{2,O,(\boldsymbol{\alpha},\alpha_{i})}_{0}(X\natural S^{1}\times B^{3};K(i),L;\eta_{i})\to \underline{\overline{\mathcal{S}}}^{2,O,\boldsymbol{\alpha}}_{0}(X;K,L;\eta) \\ &\Phi^{(\overline{\boldsymbol{\alpha}},\overline{\eta})}_{T},\Psi^{(\overline{\boldsymbol{\alpha}},\overline{\eta})}_{T}:\bigoplus_{i=1}^{m}\underline{\overline{\mathcal{S}}}^{2,T,(\overline{\boldsymbol{\alpha}},\overline{\alpha}_{i})}_{0}(X\natural S^{1}\times B^{3};K(i),L;\overline{\eta}_{i})\to \underline{\overline{\mathcal{S}}}^{2,T,\overline{\boldsymbol{\alpha}}}_{0}(X;K,L;\overline{\eta}) \end{align}\] between the corresponding cabled 1-dimensional inputs skein lasagna modules for each \((\boldsymbol{\alpha},\eta)\in H_{2}^{L'}(X';\mathbb{Z})\) and \((\overline{\boldsymbol{\alpha}},\overline{\eta})\in H_{2}^{L'}(X';\mathbb{Z}/2)\). With this in mind, we have the following theorem which calculates the 1-dimensional inputs skein lasagna modules of \((X',L')\):

As a corollary, we spell out the above theorem in the special case where \(X=\natural^{n}S^{1}\times B^{3}\) is a 1-handlebody:

The rest of this section is devoted to the proof of Theorem 16. Before we define the relevant maps, the following lemma will prove useful:

We are now ready to prove Theorem 16. For simplicity we treat the case where \(\diamond=O\); the case \(\diamond=T\) is proved similarly and is left to the interested reader. Fix a class \((\boldsymbol{\alpha},\eta)\in H_{2}^{L'}(X';\mathbb{Z})\). For each \(\mathbf{k}^{+},\mathbf{k}^{-}\in\mathbb{N}^{m}\) such that \(\mathbf{k}^{+}-\mathbf{k}^{-}=\boldsymbol{\alpha}\), we define a map \[\widetilde{f}_{O,\mathbf{k}^{\pm},\eta}:\overline{\mathcal{S}}_{0}^{2,O}\big(\mathbf{k}^{\pm};\eta_{\mathbf{k}^{\pm}}\big)\big\{-|\mathbf{k}^{+}|-|\mathbf{k}^{-}|\big\}\to\overline{\mathcal{S}}_{0}^{2,O}(X',L';(\boldsymbol{\alpha},\eta))\] as follows: let \[v=\sum_{k=1}^{N}a_{k}[F_{k}]\in\overline{\mathcal{S}}_{0}^{2,O}\big(\mathbf{k}^{\pm};\eta_{\mathbf{k}^{\pm}}\big)\] be a \(\mathbb{Q}\)-linear combination of some collection of type \(O\) 1-dimensional input lasagna fillings \(\{F_{k}\}=\{(\Sigma_{k},\{(B_{k,j},L_{k,j},v_{k,j})\})\}\) of \((X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\). For each \(k\), we have an identification \[\partial\Sigma_{k}=\sqcup_{j}(-L_{k,j})\sqcup K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L.\] By capping off the components of \(\partial\Sigma_{k}\) corresponding to the components of \(K(\mathbf{k}^{+},\mathbf{k}^{-})\) with \(k_{i}^{+}\) (respectively, \(k_{i}^{-}\)) positively-oriented (respectively, negatively oriented) core-parallel disks of the \(i\)-th 2-handle for each \(i=1,\dots, m\), we obtain a 1-dimensional input lasagna filling \(\widetilde{f}_{O,\mathbf{k}^{\pm},\eta}(F_{k})\) of \((X',L')\). We then set \(\widetilde{f}_{O,\mathbf{k}^{\pm},\eta}(v):=\sum_{k=1}^{N}a_{k}[\widetilde{f}_{O,\mathbf{k}^{\pm},\eta}(F_{k})]\). Now let \[\widetilde{f}^{(\boldsymbol{\alpha},\eta)}_{O}:=\bigoplus_{\substack{\mathbf{k}^{\pm}\in\mathbb{N}^{m} \\ \mathbf{k}^{+}-\mathbf{k}^{-}=\boldsymbol{\alpha}}}\widetilde{f}_{O,\mathbf{k}^{\pm},\eta}.\] To show that \(\widetilde{f}^{(\boldsymbol{\alpha},\eta)}_{O}\) descends to a map \[f^{(\boldsymbol{\alpha},\eta)}_{O}:\text{Coeq}(\Phi_{O}^{(\boldsymbol{\alpha},\eta)},\Psi_{O}^{(\boldsymbol{\alpha},\eta)})\to\overline{\mathcal{S}}_{0}^{2,O}(X',L';(\boldsymbol{\alpha},\eta)),\] it suffices to check that \(\widetilde{f}^{(\boldsymbol{\alpha},\eta)}_{O}\) respects the braiding, annulus, and lasso relations. The fact that \(\widetilde{f}^{(\boldsymbol{\alpha},\eta)}_{O}\) respects the braiding and annulus relations follows along the same lines as in ([5], Theorem 1.1), so we will omit the proof here.

For the lasso relation, let \(\mathbf{k}^{+},\mathbf{k}^{-}\in\mathbb{N}^{m}\) and \(\ell^{+},\ell^{-}\in\mathbb{N}\) be such that \(\mathbf{k}^{+}-\mathbf{k}^{-}=\boldsymbol{\alpha}\) and \(\ell^{+}-\ell^{-}=\alpha_{i}\). Let \(\widetilde{F}=(\widetilde{\Sigma},\{(\widetilde{B}_{j},\widetilde{L}_{j},\widetilde{v}_{j})\})\) be a type \(O\) 1-dimensional input lasagna filling of \((X\natural S^{1}\times B^{3},K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L)\). Furthermore, let \(\Phi^{(\boldsymbol{\alpha},\eta)}_{O}(F)\) (respectively, \(\Psi^{(\boldsymbol{\alpha},\eta)}_{O}(F)\)) denote the lasagna filling of \((X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\) (respectively, \((X,K(\mathbf{k}^{+}+(\ell^{+}-k_{i}^{+})\mathbf{e}_{i},\mathbf{k}^{-}+(\ell^{-}-k_{i}^{-})\mathbf{e}_{i})\cup L)\)) obtained by gluing the relative cobordism \((W_{i},\Sigma_{\mathbf{k}^{\pm},\ell^{\pm},i})\) (respectively, \((W_{i},\Sigma_{\mathbf{k}^{\pm},\ell^{\pm},i})\)) to \(\widetilde{F}\). Via the diffeotopy from Lemma 2 we obtain an isotopy from \(\widetilde{f}^{(\boldsymbol{\alpha},\eta)}_{O}(\Phi^{(\boldsymbol{\alpha},\eta)}_{O}(\widetilde{F}))\) and \(\widetilde{f}^{(\boldsymbol{\alpha},\eta)}_{O}(\Psi^{(\boldsymbol{\alpha},\eta)}_{O}(\widetilde{F}))\) as lasagna fillings of \((X',L')\). Hence \((\widetilde{f}^{(\boldsymbol{\alpha},\eta)}_{O}\circ\Phi^{(\boldsymbol{\alpha},\eta)}_{O})(v)=(\widetilde{f}^{(\boldsymbol{\alpha},\eta)}_{O}\circ\Psi^{(\boldsymbol{\alpha},\eta)}_{O})(v)\) for all \(v\in\bigoplus_{i=1}^{m}\underline{\overline{\mathcal{S}}}^{2,O,(\boldsymbol{\alpha},\alpha_{i})}_{0}(X\natural S^{1}\times B^{3};K(i),L;\eta_{i})\), as desired.

Next, we define an inverse \[(f^{(\boldsymbol{\alpha},\eta)}_{O})^{-1}:\overline{\mathcal{S}}_{0}^{2,O}(X',L';(\boldsymbol{\alpha},\eta))\to\text{Coeq}(\Phi^{(\boldsymbol{\alpha},\eta)}_{O},\Psi^{(\boldsymbol{\alpha},\eta)}_{O})\] as follows: Given a type \(O\) 1-dimensional input lasagna filling \(F\) of \((X',L')\) with surface \(\Sigma\) in skein grading \((\boldsymbol{\alpha},\eta)\), isotope the input 1-handlebodies of \(F\) to be inside \(\text{int}(X)\subset X'\), and isotope the surface \(\Sigma\) so that its intersection with the 2-handles is precisely a union of core parallel disks. By removing these 2-handles along with these core-parallel disks, we obtain a lasagna filling \((f^{(\boldsymbol{\alpha},\eta)}_{O})^{-1}(F)\) of \((X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\) for some \(\mathbf{k}^{\pm}\in\mathbb{N}^{m}\) satisfying \(\mathbf{k}^{+}-\mathbf{k}^{-}=\boldsymbol{\alpha}\).

In order to show that \((f^{(\boldsymbol{\alpha},\eta)}_{O})^{-1}\) is well-defined, let \[\{F_{t}\}_{t\in[0,1]}=\{(\Sigma_{t},\{(B_{j,t},L_{j,t},v_{j,t})\}_{j\in{J}})\}_{t\in[0,1]}\] be a 1-parameter family of type \(O\) 1-dimensional input lasagna fillings of \((X',L')\) such that for each \(k\in\{0,1\}\) the following holds:

1. \(\{B_{j,k}\}_{j\in J}\subset\text{int}(X)\subset X'\).

2. The intersection of \(\Sigma_{k}\) with the 2-handles is precisely a union of core parallel disks.

Let \(G_{1},\dots,G_{m}\) denote the co-cores of the 2-handles, so that \(\Sigma_{k}\cap{G_{i}}\) consists of \(k_{i,k}^{+}\) positively-oriented (respectfully, \(k_{i,k}^{-}\) negatively-oriented) points for some \(k_{i,k}^{\pm}\in\mathbb{N}\), for \(k=0,1\) and for each \(i\in\{1,\dots,m\}\). Define submanifolds \(Y=\bigcup_{t\in{[0,1]}}(\{t\}\times{\Sigma_{t})}\) and \(B=\bigsqcup_{j\in{J}}\bigcup_{t\in{[0,1]}}(\{t\}\times{B_{j,t}})\), then \(Y\) and \(B\) generically intersect \([0,1]\times{G_{i}}\) in a 1- and 3-manifold respectively. Deformation retracting \(B\) to its core yields a 1-parameter family of 1-complexes \(\Gamma(B)=\bigsqcup_{j\in{J}}\bigcup_{t\in{[0,1]}}(\{t\}\times{\Gamma(B_{j,t})})\) intersecting \([0,1]\times{G_{i}}\) in a finite number of points (see Remark 10). After a possible perturbation of \(\Gamma(B)\) and \(Y\) rel \(\partial\), we may assume that \(\Gamma_{i}:=(Y\cup \Gamma(B))\cap([0,1]\times{G_{i}})\) is an embedded graph in \([0,1]\times G_{i}\approx[0,1]\times D^{2}\) such that if \(\pi_{i}\) denotes the composition \[\Gamma_{i}\hookrightarrow{[0,1]\times G_{i}}\twoheadrightarrow[0,1]\] of the inclusion map followed by projection onto the first factor, then \(\pi_{i}\) is a local diffeomorphism outside of finitely many critical points of the forms depicted in Figure 7.

Similar to ([5], Proof of Theorem 1.1), in the intervals between the critical points, the intersections between \(\Sigma_{t}\) and \(G_{i}\) change by an isotopy of \(G_{i}\) and hence correspond to the braid group action, and passing through the cup and cap critical points correspond to the \(\psi_{O,i}^{[j]}\) maps. In the 1-dimensional inputs setting, we also have the appearance of switch critical points, which correspond to pieces of 1-dimensional inputs passing through \(G_{i}\).

The topology of the 1-dimensional inputs \(\{B_{j,t}\}\) and surfaces \(\{\Sigma_{j,t}\}\) may apriori be arbitrarily complicated. However we have the following lemma which allows us to greatly simplify the topology of the situation in the presence of a switch point:

4.5 Properties of \(\Phi\) and \(\Psi\)↩︎

In this section we establish some properties of the maps \(\Phi_{\diamond}^{(\boldsymbol{\alpha},\eta)}\) and \(\Psi_{\diamond}^{(\boldsymbol{\alpha},\eta)}\) from Theorem 16, with components defined by (12 ) and (13 ), respectively.

Let \(i\in\{1,\dots,m\}\), and fix \(\mathbf{k}^{+},\mathbf{k}^{-}\in\mathbb{N}^{m}\), \(\ell^{+},\ell^{-}\in\mathbb{N}\) such that

• \(k_{i}^{+}-k_{i}^{-}=\ell^{+}-\ell^{-}\) if \(\diamond=O\),

• \(k_{i}^{+}-k_{i}^{-}\equiv \ell^{+}-\ell^{-}\pmod{2}\) if \(\diamond=T\).

First, note that the decompositions (10 ) and (11 ) give us factorizations \[\begin{align} &\Phi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}=\overline{\mathcal{S}}_{0}^{2,\diamond}(Y\times[0,1],C_{\ell^{\pm}})\circ\overline{\mathcal{S}}_{0}^{2,\diamond}(Z_{i},S_{\mathbf{k}^{\pm},\ell^{\pm},i}), \\ &\Psi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}=\overline{\mathcal{S}}_{0}^{2,\diamond}(Y\times[0,1],C_{k_{i}^{\pm}})\circ\overline{\mathcal{S}}_{0}^{2,\diamond}(Z'_{i},S'_{\mathbf{k}^{\pm},\ell^{\pm},i}). \end{align}\] First, we calculate the maps \(\overline{\mathcal{S}}_{0}^{2,\diamond}(Z_{i},S_{\mathbf{k}^{\pm},\ell^{\pm},i})\) and \(\overline{\mathcal{S}}_{0}^{2,\diamond}(Z'_{i},S'_{\mathbf{k}^{\pm},\ell^{\pm},i})\). Observe that \(Z_{i}\) is given by an elementary cancelling 2-handle attachment, i.e., a 2-handle attachment along a longitude in an \(S^{1}\times S^{2}\)-summand. In the special case where \(X=\natural^{n}S^{1}\times B^{3}\), the cobordism \((Z_{i},S_{\mathbf{k}^{\pm},\ell^{\pm},i})\) corresponds to the elementary morphism (v) of ([10], Proposition 6.3). Hence \[\label{eq:Z95i95equals95counit} \overline{\mathcal{S}}_{0}^{2,\diamond}(Z_{i},S_{\mathbf{k}^{\pm},\ell^{\pm},i})=\mathrm{KhR}_{2,\diamond}^{-}(Z_{i},S_{\mathbf{k}^{\pm},\ell^{\pm},i})=\epsilon\tag{16}\] where \[\label{eq:counit95iota} \epsilon:\mathrm{KhR}_{2,\diamond}^{-}\big(K(\mathbf{k}^{\pm};\ell^{\pm};i)\big)\to\mathrm{KhR}_{2,\diamond}^{-}\big(K(\mathbf{k}^{+},\mathbf{k}^{-})\big)\otimes\mathrm{KhR}_{2}\big(U(\ell^{+},\ell^{-})\big)\tag{17}\] is the counit map which deletes the projector (see Remark 1).

Next, note that the 2-handle cobordism \((Z'_{i},S'_{\mathbf{k}^{\pm},\ell^{\pm},i})\) is (in general) not an elementary cancelling 2-handle attachment. However, we can decompose it as a composition of a diffeomorphism of the boundary manifold/link pair followed by an elementary canceling 2-handle attachment.

From Lemma 4 we immediately obtain a diffeomorphism of pairs \[\varphi_{\mathbf{k}^{\pm},\ell^{\pm},i}:\big(Y\#S^{1}\times S^{2},K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L\big)\xrightarrow{\approx} \big(Y\#S^{1}\times S^{2},K(\mathbf{k}^{\pm}+(\ell^{\pm}-k_{i}^{\pm})\mathbf{e}_{i};k_{i}^{\pm};i)\cup L\big)\] which sends:

• \(K(i)_{i}(k_{i}^{+},k_{i}^{-})\) to \(K(i)_{m+1}(k_{i}^{+},k_{i}^{-})\),

• \(K(i)_{m+1}(\ell^{+},\ell^{-})\) to \(K(i)_{i}(\ell^{+},\ell^{-})\), and

• every other component of \(K(\mathbf{k}^{\pm};\ell^{\pm};i)\cup L\) to themselves,

and which is well-defined up to composition with a diffeomorphism of pairs which braids cable components.

For the moment, let us specialize to the case where \(X=\natural^{n}S^{1}\times B^{3}\). By (16 ) and Proposition 17 we have that \[\label{eq:Z3995i95connected95sums95S951xS952} \mathrm{KhR}_{2,\diamond}^{-}(Z'_{i},S'_{\mathbf{k}^{\pm},\ell^{\pm},i})=\epsilon\circ\mathrm{KhR}_{2,\diamond}^{-}(\varphi_{\mathbf{k}^{\pm},\ell^{\pm},i}).\tag{19}\]

In order to compute the map \(\mathrm{KhR}_{2,\diamond}^{-}(\varphi_{\mathbf{k}^{\pm},\ell^{\pm},i})\) in practice, i.e., using diagrammatics, it will be helpful to be able to replace \(\varphi_{\mathbf{k}^{\pm},\ell^{\pm},i}\) instead with a diffeomorphism of pairs \(\psi_{\mathbf{k}^{\pm},\ell^{\pm},i}\) satisfying the same abstract properties as \(\varphi_{\mathbf{k}^{\pm},\ell^{\pm},i}\). Before spelling out the necessary conditions on \(\psi_{\mathbf{k}^{\pm},\ell^{\pm},i}\), the following lemma will be useful:

We now return to our task at hand:

We can therefore replace (19 ) with the equation \[\label{eq:Z3995i95connected95sums95S951xS95295redux} \mathrm{KhR}_{2,\diamond}^{-}(Z'_{i},S'_{\mathbf{k}^{\pm},\ell^{\pm},i})=\epsilon\circ\mathrm{KhR}_{2,\diamond}^{-}(\psi_{\mathbf{k}^{\pm},\ell^{\pm},i}).\tag{20}\] where \(\psi_{\mathbf{k}^{\pm},\ell^{\pm},i}\) is as in Proposition 18.

We now return to the case of more general \(X\), and calculate the map \(\overline{\mathcal{S}}_{0}^{2,\diamond}(Y\times[0,1],C_{\ell^{\pm}})\). Consider the identification \[\overline{\mathcal{S}}_{0}^{2,\diamond}\big(X;K(\mathbf{k}^{+},\mathbf{k}^{-})\cup U(\ell^{+},\ell^{-})\cup L\big)\cong\overline{\mathcal{S}}_{0}^{2,\diamond}\big(X;K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\big)\otimes\mathrm{KhR}_{2}\big(U(\ell^{+},\ell^{-})\big)\] induced by the identification of pairs \[\big(X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup U(\ell^{+},\ell^{-})\cup L\big)\cong \big(X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\big)\natural\big(B^{4},U(\ell^{+},\ell^{-})\big)\] and the canonical isomorphism \[\overline{\mathcal{S}}_{0}^{2,\diamond}\big(B^{4};U(\ell^{+},\ell^{-})\big)\cong\mathrm{KhR}_{2}\big(U(\ell^{+},\ell^{-})\big).\] Then we can write \[\begin{align} \overline{\mathcal{S}}_{0}^{2,\diamond}(Y\times[0,1],C_{\ell^{\pm}})=\text{id}\otimes\varepsilon_{\ell^{\pm}}:\overline{\mathcal{S}}_{0}^{2,\diamond}\big(X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\big)\otimes&\mathrm{KhR}_{2}\big(U(\ell^{+},\ell^{-})\big) \\ &\to\overline{\mathcal{S}}_{0}^{2,\diamond}\big(X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\big) \end{align}\] where \[\label{eq:counit95epsilon} \varepsilon_{\ell^{\pm}}:\mathrm{KhR}_{2}\big(U(\ell^{+},\ell^{-})\big)\to\mathbb{Q}\tag{21}\] is the counit map associated to the cobordism which caps off all of the components with disks. More concretely, \(\varepsilon_{\ell^{\pm}}\) factors as \[\mathrm{KhR}_{2}\big(U(\ell^{+},\ell^{-})\big)\xrightarrow{\cong}(\mathrm{KhR}_{2}(U))^{\otimes(\ell^{+}+\ell^{-})}\xrightarrow{\varepsilon^{\otimes(\ell^{+}+\ell^{-})}}\mathbb{Q}\] where \(\varepsilon:\mathrm{KhR}_{2}(U)\to\mathbb{Q}\) is the (Frobenius) counit map which sends \(\mathbf{1}\mapsto 0\) and \(\mathbf{X}\mapsto 1\). Similarly we have that \[\overline{\mathcal{S}}_{0}^{2,\diamond}(Y\times[0,1],C_{k_{i}^{\pm}})=\text{id}\otimes\varepsilon_{k_{i}^{\pm}}.\] Putting this all together, in the case where \(X=\natural^{n}S^{1}\times B^{3}\) we can write \[\begin{align} \label{eq:Phi95Psi95decomps95195hbody} &\Phi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}=( \mathrm{id}\otimes\varepsilon_{\ell^{\pm}})\circ\epsilon, & &\Psi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}=( \mathrm{id}\otimes\varepsilon_{k_{i}^{\pm}})\circ\epsilon\circ\mathrm{KhR}_{2,\diamond}^{-}(\psi_{\mathbf{k}^{\pm},\ell^{\pm},i}) \end{align}\tag{22}\] where \(\psi_{\mathbf{k}^{\pm},\ell^{\pm},i}\) is as in Proposition 18.

It will be helpful to establish some further algebraic properties of the collections of maps \(\{\Phi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}\}\) and \(\{\Psi_{\diamond,\mathbf{k}^{\pm},\ell^{\pm},i}\}\). The following proposition posits a symmetry property between the \(\Phi\) and \(\Psi\) maps:

Next, we show that the lasso relation subsumes the annulus relations:

Next, we specialize to the \(\diamond=T\) theory. In this setting, the lasso relation includes two new sets of relations, which we call the disoriented annulus relations. Let \(i\in\{1,\dots,m\}\), let \(\mathbf{k}^{\pm}\in\mathbb{N}^{m}\), and let \(B\) be a 4-dimensional tubular neighborhood of a push off of \(K_{i}\times\{\frac{1}{2}\}\subset Y\times[0,1]\). Let \(\widetilde{\mathbb{1}}({2,0})\subset\partial B\cong S^{1}\times S^{2}\) denote the belt link with two positively oriented strands, and consider the surface \(\Sigma_{+}\) in \(Y\times[0,1]\setminus B\) with boundary \[\partial\Sigma_{+}=\big(-(K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\times\{0\}\big)\sqcup\big(-\widetilde{\mathbb{1}}({2,0})\big)\sqcup\big((K(\mathbf{k}^{+}+2\mathbf{e}_{i},\mathbf{k}^{-})\cup L)\times\{1\}\big)\] which consists of:

• trivial cylinders connecting the components of \(K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\subset Y\times\{0\}\) to their corresponding components in the sublink \(K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L\subset K(\mathbf{k}^{+}+2\mathbf{e}_{i},\mathbf{k}^{-})\cup L\subset Y\times\{1\}\).

• two cylinders parallel to \(K_{i}\times[\frac{1}{2},1]\) connecting \(\widetilde{\mathbb{1}}({2,0})\subset\partial B\) to the remaining two positively-oriented components of \(K(\mathbf{k}^{+}+2\mathbf{e}_{i},\mathbf{k}^{-})\cup L\subset Y\times\{1\}\).

For each \(u\in\mathrm{KhR}_{2,T}^{-}(\widetilde{\mathbb{1}}({2,0}))\) consider the type \(T\) 1-dimensional inputs lasagna filling \[F^{+}_{u}:=(\Sigma_{+},(B,\widetilde{\mathbb{1}}({2,0}),v))\] of the pair \[\big(Y\times[0,1],-(K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\sqcup(K(\mathbf{k}^{+}+2\mathbf{e}_{i},\mathbf{k}^{-})\cup L)\big).\] Similar to the proof of Proposition 20, we can glue this lasagna filling to the boundary of a lasagna filling of \((X,K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\) to obtain a lasagna filling of \((X,K(\mathbf{k}^{+}+2\mathbf{e}_{i},\mathbf{k}^{-})\cup L)\), defining a map \[\psi^{+,u}_{T,i}:\overline{\mathcal{S}}_{0}^{2,T}(X;K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\to\overline{\mathcal{S}}_{0}^{2,T}(X;K(\mathbf{k}^{+}+2\mathbf{e}_{i},\mathbf{k}^{-})\cup L)\] for each \(u\in\mathrm{KhR}_{2,T}^{-}(\widetilde{\mathbb{1}}({2,0}))\) with grading shift equal to the bigrading of \(u\). By replacing \(\widetilde{\mathbb{1}}({2,0})\) with \(\widetilde{\mathbb{1}}({0,2})\) in the above discussion, we can similarly define a map \[\psi^{-,u}_{T,i}:\overline{\mathcal{S}}_{0}^{2,T}(X;K(\mathbf{k}^{+},\mathbf{k}^{-})\cup L)\to\overline{\mathcal{S}}_{0}^{2,T}(X;K(\mathbf{k}^{+},\mathbf{k}^{-}+2\mathbf{e}_{i})\cup L)\] for each \(u\in\mathrm{KhR}_{2,T}^{-}(\widetilde{\mathbb{1}}({0,2}))\), again with grading shift equal to the bigrading of \(u\).

Next, consider the maps \(\varepsilon_{+}\) and \(\varepsilon_{-}\) obtained as the compositions of the maps \[\begin{align} &\mathrm{KhR}_{2,T}^{-}\big(\widetilde{\mathbb{1}}({2,0})\big)\xrightarrow{\epsilon}\mathrm{KhR}_{2}\big(U(2,0)\big)\xrightarrow{\varepsilon^{\otimes 2}}\mathbb{Q}, \\ &\mathrm{KhR}_{2,T}^{-}\big(\widetilde{\mathbb{1}}({0,2})\big)\xrightarrow{\epsilon}\mathrm{KhR}_{2}\big(U(0,2)\big)\xrightarrow{\varepsilon^{\otimes 2}}\mathbb{Q}, \end{align}\] respectively. The proof of the following proposition is very similar to that of Proposition 20, so we leave it as an exercise to the motivated reader:

One can show (e.g., by comparing with the corresponding calculation in Section 4.1 in [12]) that the maps \(\varepsilon_{\pm}\) vanish in bidegrees \(\neq (0,2)\), and in bigrading \((0,2)\) the maps \(\varepsilon_{\pm}\) induce isomorphisms \[\begin{align} &\mathrm{KhR}_{2,T}^{-,0,2}\big(\widetilde{\mathbb{1}}({2,0})\big)\underset{\cong}{\xrightarrow{\varepsilon_{+}}}\mathbb{Q}, & &\mathrm{KhR}_{2,T}^{-,0,2}\big(\widetilde{\mathbb{1}}({0,2})\big)\underset{\cong}{\xrightarrow{\varepsilon_{-}}}\mathbb{Q}. \end{align}\] Define \(\mathbf{X}_{+}\in\mathrm{KhR}_{2,T}^{-,0,2}(\widetilde{\mathbb{1}}({2,0}))\) and \(\mathbf{X}_{-}\in\mathrm{KhR}_{2,T}^{-,0,2}(\widetilde{\mathbb{1}}({0,2}))\), respectively, as the preimages of \(1\in\mathbb{Q}\) under the above isomorphisms. Proposition 21 then implies the relations \[\begin{align} \label{eq:disoriented95annulus95relations} &\psi_{T,i}^{+,u_{+}}(v)\sim\psi_{T,i}^{-,u_{-}}(v)\sim 0, & &\psi_{T,i}^{+,\mathbf{X}_{+}}(v)\sim\psi_{T,i}^{-,\mathbf{X}_{-}}(v)\sim v, \end{align}\tag{23}\] in the coequalizer from Theorem 16 for all \(u_{+}\in\mathrm{KhR}_{2,T}^{-,h,q}(\widetilde{\mathbb{1}}({2,0}))\) and \(u_{-}\in\mathrm{KhR}_{2,T}^{-,h,q}(\widetilde{\mathbb{1}}({0,2}))\) such that \((h,q)\neq(0,2)\).

5 1-, 3-, & 4-Handles↩︎

In this section we provide formulae for handle attachments of index \(\neq{2}\). Let \(X\) admit a handle decomposition \(X_{1}\subset{X_{2}}\subset{X_{3}}\subset{X_{4}}=X\) where \(X_{i+1}\) is obtained from \(X_{i}\) by attaching the index \((i+1)\) handles of \(X\). Let \(L\) denote a link in the boundary \(\partial{X_{1}}\), and let \(K\) denote the framed attaching link of the 2-handles of \(X_{2}\). For the pair \((X_{1},L)\), and in analogy with the 0-dimensional inputs invariant evaluated on the pair \((B^{4},L)\), the 1-dimensional inputs skein lasagna module recovers the Rozansky-Willis homology of \(L\subset{\sqcup^{m}\#^{n}S^{1}\times{S^{2}}}\) on pairs \((X_{1},L)\) where \(\partial{X_{1}}=\sqcup^{m}\#^{n}S^{1}\times{S^{2}}\). See ([10], discussion before Remark 3.5): As previously stated there is a natural isomorphism \[\label{eq:tortellini95195h-body} \overline{\mathcal{S}}_{0}^{2,\diamond}(\textstyle{\natural^{n}}S^{1}\times B^{3};L)\cong\mathrm{KhR}_{2,\diamond}^{-}(\#^{n}S^{1}\times{S^{2}};L)\tag{24}\] given by \([(L\times{I},B,v)]\mapsto{v}\), where \((L\times{I},B,v)\) is the 1-dimensional inputs filling with input manifold \(B\) diffeomorphic to a smaller copy of \(X_{1}\) embedded in the original, and \(L\times{I}\) the product cobordism in the relative handlebody \(X_{1}\setminus{\text{int}(B)}\).

We are now ready to prove Theorem 1:

6 Calculations↩︎

In this section we use Corollary 2 to compute \(\overline{\mathcal{S}}^{2,\diamond}_{0}(X;L)\) for various \(4\)-manifolds \(X\) built out of \(1\)-and \(2\)-handles.

6.1 \(\mathbf{S^{2}\times D^{2}}\)↩︎

Recall that \(S^{2}\times D^{2}\) can be constructed by attaching a 2-handle along the zero-framed unknot \(U=(U,0)\subset\partial B^{4}\). Fix \(\alpha\in\mathbb{Z}\), and let \(r,s\in\mathbb{N}\). Note that \(U(1)\)³ can be identified with the belt link \(\widetilde{\mathbb{1}}({1,1})\subset S^{1}\times S^{2}\), and hence we have the following identification: \[U(\alpha^{\pm}+r;\alpha^{\pm}+s;1)=\widetilde{\mathbb{1}}({\alpha^{+}+r,\alpha^{+}+r})\cup\widetilde{\mathbb{1}}({\alpha^{-}+s,\alpha^{+}+s})\subset S^{1}\times S^{2}.\] From (22 ) we have that \[\begin{align} &\Phi_{\diamond,\alpha^{\pm}+r,\alpha^{\pm}+s,1}=(\text{id}\otimes\varepsilon_{\alpha^{\pm}+s})\circ\epsilon, & &\Psi_{\diamond,\alpha^{\pm}+r,\alpha^{\pm}+s,1}=( \mathrm{id}\otimes\varepsilon_{\alpha^{\pm}+r})\circ\epsilon\circ\mathrm{KhR}_{2,\diamond}^{-}(\psi_{\alpha^{\pm}+r,\alpha^{\pm}+s,1}). \end{align}\] where \(\psi_{\alpha^{\pm}+r,\alpha^{\pm}+s,1}\) is as in Example 1. Observe that we can rewrite the map \(\Psi_{\diamond,\alpha^{\pm}+r,\alpha^{\pm}+s,1}\) as \[\Psi_{\diamond,\alpha^{\pm}+r,\alpha^{\pm}+s,1}=(\varepsilon_{\alpha^{\pm}+r}\otimes \mathrm{id})\circ( \mathrm{id}\otimes g)\circ\epsilon.\] where \[g:\mathrm{KhR}_{2}\big(U(\alpha^{-}+s,\alpha^{+}+s)\big)\{-|\alpha|-2s\}\xrightarrow{\cong}\mathrm{KhR}_{2}\big(U(\alpha^{+}+s,\alpha^{-}+s)\big)\{-|\alpha|-2s\}\] is the canonical isomorphism given by \[\begin{align} &\mathcal{V}^{\otimes(\alpha^{-}+s)}\otimes\mathcal{V}^{\otimes(\alpha^{+}+s)}\xrightarrow{\cong}\mathcal{V}^{\otimes(\alpha^{+}+s)}\otimes\mathcal{V}^{\otimes(\alpha^{-}+s)}, & &\mathcal{V}:=\mathrm{KhR}_{2}(U)\{-1\}. \end{align}\] From Corollary 2 and Proposition 20 it follows that \[\overline{\mathcal{S}}^{2,O}_{0}(S^{2}\times D^{2};\alpha)\cong\Big(\bigoplus_{r=0}^{\infty}\mathrm{KhR}_{2}\big(U(\alpha^{+}+r,\alpha^{-}+r)\big)^{\mathfrak{S}_{\alpha^{\pm}+r}}\{-|\alpha|-2r\}\Big)/\sim\] where \(\sim\) is the transitive and linear closure of the relations \[\big[( \mathrm{id}\otimes\varepsilon_{\alpha^{\pm}+s})(v)\big]\sim\big[(\varepsilon_{\alpha^{\pm}+r}\otimes \mathrm{id})(v)\big]\] for all \(r,s\in\mathbb{N}\) and all \(v\in\text{im}(F_{\alpha^{\pm}+r,\alpha^{\pm}+s})\), where \[F_{\alpha^{\pm}+r,\alpha^{\pm}+s}:=(\text{id}\otimes g)\circ\epsilon.\] Finally, note that \(\overline{\mathcal{S}}^{2,\diamond}_{0}(S^{2}\times D^{2})\) can be given the structure of a \(\mathbb{Q}\)-algebra via the gluing \[S^{2}\times D^{2}\cup_{S^{2}\times I}S^{2}\times D^{2}\approx S^{2}\times D^{2}\] as in the case of the ordinary skein lasagna module (see [5], Section 5).

We have the following corollary:

We also have the following analogue of ([10], Theorem 2.7):

6.2 \(\mathbf{D(p)}\), \(p\neq 0\)↩︎

Let \(D(p)\) denote the \(D^{2}\)-bundle over \(S^{2}\) with Euler number \(p\in\mathbb{Z}\). As calculated in ([5], Theorem 1.3), we have that \[\label{eq:D40p4195lasagna95vanishing95p620} \mathcal{S}_{0,0,*}^{2}(D(p);0)=0\tag{26}\] for \(p>0\), and for \(p<0\) it was shown that \[\label{eq:D40p4195lasagna95p600} \mathcal{S}_{0,0,*}^{2}(D(p);0)\cong\mathbb{Q}\tag{27}\] concentrated in \(q\)-degree \(0\). One can say more about the former case. Indeed, ([2], Theorem 1.4) implies that \(\mathcal{S}_{0}^{2}(D(p))\) vanishes in all tri-gradings. As mentioned in the introduction, these same statements hold for \(1\)-dimensional inputs lasagna:

We will dedicate the remainder of this section to proving Proposition 25. The first statement follows from (26 ) and Proposition 14. For the rest of this section, let \(p<0\). In order to prove the second statement, it suffices to show that the surjection \[f_{0,\diamond}:\mathcal{S}_{0,0,0}^{2}(D(p);0)\twoheadrightarrow\overline{\mathcal{S}}_{0,0,0}^{2,\diamond}(D(p);0)\] from Proposition 12 in tri-degree \((0,0,0)\) is an isomorphism for \(\diamond\in\{O,T\}\). Let \((U,p)\) denote the \(p\)-framed unknot, and note that a \((k^{+},k^{-})\)-cable of \((U,p)\) is given by \[(U,p)(k^{+},k^{-})=T(k^{+}+k^{-},p(k^{+}+k^{-}))_{k^{+},k^{-}},\] where \(T(k^{+}+k^{-},p(k^{+}+k^{-}))_{k^{+},k^{-}}\) denotes the corresponding torus link, with each component having framing \(p\) and with the subscripts denoting that \(k^{+}\) of the strands are oriented positively, and \(k^{-}\) of the strands are oriented negatively. Next, let \(\widetilde{\mathbb{1}}({k^{+},k^{-}})_{p}\subset S^{1}\times S^{2}\) denote the \(p\)-twisted standard belt link, as described in Section 2.1. Note that we can identify \[(U,p)(1)=\widetilde{\mathbb{1}}({1,0})_{p}\cup\widetilde{\mathbb{1}}({0,1})\subset S^{1}\times S^{2},\] and so consequently for any \(k^{\pm},\ell^{\pm}\in\mathbb{N}\) we have an identification \[(U,p)(k^{\pm};\ell^{\pm};1)=\widetilde{\mathbb{1}}({k^{+},k^{-}})_{p}\cup\widetilde{\mathbb{1}}({\ell^{+},\ell^{-}})\subset S^{1}\times S^{2}.\] As in (22 ) we can identify \(\Phi_{\diamond,k^{\pm},\ell^{\pm},1}\) and \(\Psi_{\diamond,k^{\pm},\ell^{\pm},1}\) with the compositions of maps as pictured in Figure 16.

For all \(k^{+},k^{-}\in\mathbb{N}\) such that

• \(k^{+}=k^{-}\) if \(\diamond=O\),

• \(k^{+}\equiv k^{-}\pmod{2}\) if \(\diamond=T\),

we define \(v_{\diamond,k^{\pm},p}\in\mathrm{KhR}_{2,\diamond}^{0,k^{+}+k^{-}}(T(k^{+}+k^{-},p(k^{+}+k^{-}))_{k^{+},k^{-}})\) to be the element \[\label{eq:v95diamond44k95pm95p} v_{\diamond,k^{\pm},p}:= { \left\{ \begin{array}{ll} \underbrace{(\psi^{[1]}\circ\cdots\circ\psi^{[1]})}_{k}(1) & ifk^{+}=k^{-}=:k, \\ \underbrace{(\psi^{[1]}\circ\cdots\circ\psi^{[1]})}_{k^{-}}\underbrace{(\psi_{T}^{+,\mathbf{X}_{+}}\circ\cdots\circ\psi_{T}^{+,\mathbf{X}_{+}})}_{\frac{1}{2}(k^{+}-k^{-})}(1) & ifk^{+}>k^{-}, \\ \underbrace{(\psi^{[1]}\circ\cdots\circ\psi^{[1]})}_{k^{+}}\underbrace{(\psi_{T}^{-,\mathbf{X}_{-}}\circ\cdots\circ\psi_{T}^{-,\mathbf{X}_{-}})}_{\frac{1}{2}(k^{-}-k^{+})}(1) & ifk^{+}<k^{-}, \end{array} \right. }\tag{28}\] where \(1\in\mathrm{KhR}_{2}^{0,0}(S^{3},\emptyset)\cong\mathbb{Q}\) and \(\psi_{T}^{\pm,\mathbf{X}_{\pm}}\) are the disoriented annulus maps from Proposition 21 (note that the last two cases only occur when \(\diamond=T\)). We claim that \(\overline{\mathcal{S}}_{0,0,0}^{2,\diamond}(D(p);0)\cong\mathbb{Q}\) is generated by the images of the elements \(v_{T,k^{\pm},p}\), with each element (after fixing a global sign) corresponding to \(1\in\mathbb{Q}\).

6.3 \(\mathbf{B^{4}}\) {#sec:sec:subsec:B^{4}}↩︎

By 24 we have an isomorphism \[\overline{\mathcal{S}}^{2,\diamond}_{0}(B^{4})\cong\mathrm{KhR}_{2,\diamond}^{-}(S^{3},\emptyset)\cong\mathbb{Q}\] concentrated in tri-degree \((0,0,0)\) for \(\diamond\in\{O,T\}\). In this section we will (re-)calculate \(\overline{\mathcal{S}}^{2,\diamond}_{0}(B^{4};\emptyset)\) using the Kirby diagram of \(B^{4}\) with a single 1-handle and a \(p\)-framed canceling 2-handle, as pictured in Figure 21.

First consider the case where \(\diamond=O\). Using the canceling 1- and 2-handle pair description of \(B^{4}\), Corollary 2 furnishes an isomorphism \[\overline{\mathcal{S}}^{2,\diamond}_{0}(B^{4};\emptyset)\cong\Big(\bigoplus_{r=0}^{\infty}\mathrm{KhR}_{2,O}^{-}\big(\widetilde{\mathbb{1}}({r,r})_{p})\big)\{-2r\}\Big)/\sim,\] where \(\sim\) denotes the equivalence relation generated by the symmetric, bundt cake, and lasso relations. Note that we can identify \(\Phi_{O,r,s,1}\) and \(\Psi_{O,r,s,1}\) with the compositions of maps as pictured in Figure 17, and where \(\psi_{r,s,1}\) is the diffeomorphism pictured in Figure 18.

We can simplify these maps as follows. Let \(L_{r,s,p}\) be the link featured in the upper left hand corner of Figure 17, and consider the isomorphism \[\begin{align} &\mathrm{KhR}_{2,O}^{-}\big(\#^{2}S^{1}\times S^{2},L_{r,s,p}\big)\{-2r-2s\} \\ &\qquad\qquad\underset{\cong}{\xrightarrow{f}}\mathrm{KhR}_{2,O}^{-}\big(S^{1}\times S^{2},\widetilde{\mathbb{1}}({r,r})\big)\{-2r\}\otimes\mathrm{KhR}_{2,O}^{-}\big(S^{1}\times S^{2},\widetilde{\mathbb{1}}({s,s})\big)\{-2s\} \end{align}\] obtained as the composition of isomorphisms as in Figure 19. We then have that \[\begin{align} &\Phi_{O,r,s,1}=\Phi_{O,r,s,1}'\circ f & &\Psi_{O,r,s,1}=\Psi_{O,r,s,1}'\circ f \end{align}\] where \(\Phi_{O,r,s,1}'\) and \(\Psi_{O,r,s,1}'\) are the maps pictured in Figure 20.

We claim that under the above isomorphism, \(\overline{\mathcal{S}}^{2,O}_{0}(B^{4};\emptyset)\) can be identified with the one-dimensional \(\mathbb{Q}\)-vector space generated by the sequence of elements \[\widetilde{v}_{O,r,p}:=(\epsilon)^{-1}(v_{O,r,p})\in\mathrm{KhR}_{2,O}^{-,0,2r}\big(S^{1}\times S^{2},\widetilde{\mathbb{1}}({r,r})_{p}\big),\qquad\qquad r\geq 0,\] where \(v_{O,r,p}\in\mathrm{KhR}_{2}^{0,2r}(T(2r,2rp)_{r,r})\) is defined as in (28 ). Before we proceed, we shall consider the following lemma.

In view of Lemma 7, it suffices to show the following:

1. For every \(r\in\mathbb{N}\) and \(x\in\ker(\varepsilon'_{O,r,0})\), there exists some \(s\in\mathbb{N}\) and some element \[v\in\mathrm{KhR}_{2,O}^{-,0,2r}\big(S^{1}\times S^{2},\widetilde{\mathbb{1}}({r,r})\big)\otimes\mathrm{KhR}_{2,O}^{-,0,2s}\big(S^{1}\times S^{2},\widetilde{\mathbb{1}}({s,s})\big)\] such that \(\Phi'_{O,r,s,1}(v)=x\) and \(\Psi'_{O,r,s,1}(v)=0\).

2. For any \(r,s\in\mathbb{N}\) and any element \[w\in\mathrm{KhR}_{2,O}^{-}\big(S^{1}\times S^{2},\widetilde{\mathbb{1}}({r,r})\big)\{-2r\}\otimes\mathrm{KhR}_{2,O}^{-}\big(S^{1}\times S^{2},\widetilde{\mathbb{1}}({s,s})\big)\{-2s\}\] the following implication holds: \[\label{eq:w95B944} \Phi'_{O,r,s,1}(w)=\widetilde{v}_{O,r,p}\implies\Psi'_{O,r,s,1}(w)=\widetilde{v}_{O,s,p}+x\tag{34}\] for some \(x\in\ker(\varepsilon'_{O,r,0})\).

For (1), one can check that \(v=x\otimes\widetilde{v}_{O,r,0}\) satisfies the desired property. For (2), note that if \(w\) is such that \(\Phi'_{O,r,s,1}(w)=\widetilde{v}_{O,r,p}\), then we must have \(w=\widetilde{v}_{O,r,0}\otimes\widetilde{v}_{O,s,0}\), hence (34 ) holds with \(x=0\).

6.4 \(\mathbf{\Sigma_{g}\times D^{2}}\)↩︎

By Theorem 16 and observing that the Rozansky projector is supported in non-positive homological degree, one may immediately obtain \(\overline{\mathcal{S}}_{0}^{2,\diamond}\) vanishing results for manifolds represented by Kirby diagrams satisfying restrictive properties.

We remark that after fixing a cabling pattern of the link \(L\subset{\#^{n}S^{1}\times{S^{2}}}\), one may use full-twist approximation isomorphisms to compute the Rozansky-Willis homology groups that appear in the cabling colimit. For example, if we consider again the case where the 2-handle attaching knot in the Kirby diagram is 0-framed and crossingless and fix a small \(h\), then there is a corresponding fixed \(p\) satisfying 36 for all diagrams of cables \(D_{L(r,r)}\). Let \(D_{L(r,r)(\overrightarrow{p})}\) denote the diagram obtained from a diagram of \(L(r,r)\) after erasing the \(0\)-framed unknots corresponding to \(S^{1}\times{S^{2}}\) connect summands and replacing them with \(p\)-full twist braids. There are then isomorphisms of the form \[\label{eq:approxmap} \phi_{(r,r)}:\mathrm{KhR}_{2}^{h,-}(\#^{n}S^{1}\times{S^{2}};L(r,r))\rightarrow{\mathrm{KhR}_{2}^{h,-}(S^{3};L^{k}(r,r)})\tag{38}\] for each \(r\). Note that if either assumption on the attaching knot is dropped then \(k\) cannot remain fixed as the number of positive crossings in each cable diagram increases as \(r\rightarrow{\infty}\). For the maps \(\phi_{(r,r)}\) to assemble to an isomorphism between 1-dimensional inputs skein lasagna modules, one would need establish an analogue of 37 for projectors in the \(\mathfrak{gl}_{2}\) webs and foams formalism, as well as verify cobordism naturality for the maps 38 . We leave this for future work (See Questions 1,2).

References↩︎

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1. skein tortellini module, for short.↩︎

2. Bundt cake and Bundt cake with a raisin.↩︎

3. not to be confused with the other \(U(1)\).↩︎