1 Introduction↩︎

The Alexander polynomial, introduced nearly a century ago, continues to be a fundamental knot invariant. Classical approaches introduce it via the Seifert matrix, using Fox calculus, or through a presentation of the knot group and its abelianization [1], all encoding the properties of the first homology group of the infinite cyclic cover of the knot complement. In addition, several generalizations have been developed in the last decades, leading to interesting and fruitful theories [2], [3]. Other powerful knot invariants, such as knot Floer homology [4], recover it as a graded Euler characteristic, showing once again its importance and ubiquity.

There are several approaches to categorifying the Alexander polynomial, many of which give rise to quantum field theories for Alexander invariants [5]–[9]. From a representation-theoretic perspective, [10] established their deep connection with complex affine representations of the knot group, a relation first observed by De Rham [11] and Burde [12], and later studied from a sheaf-theoretic perspective by Hironaka [13]. For any knot \(K\), the space of \({\mathrm{AGL}_{1}}(\mathbb{C})\)-representations of the knot group \(\Gamma_K\) can be interpreted as a singular vector bundle (a geometric space associated to a coherent sheaf), if one considers the regular map between representation varieties \[\label{eqn:Alexfibration} \pi_K: {\mathcal{R}}_{{\mathrm{AGL}_{1}}(\mathbb{C})}(\Gamma_K) \longrightarrow {\mathcal{R}}_{{\mathbb{C}}^{\ast}}(\Gamma_K) = {\mathbb{C}}^*\tag{1}\] induced by the projection onto the diagonal entry \({\mathrm{AGL}_{1}}(\mathbb{C})\rightarrow {\mathbb{C}}^*\). The fibre of this map over any \(t\in {\mathbb{C}}^*\) can be identified with \[{\mathcal{R}}_{{\mathrm{AQ}_{t}}}(Q_K) = \mathop{\mathrm{Hom\,}}_\boldsymbol{\mathrm{Qdl}}(Q_K, {\mathrm{AQ}_{t}}),\] the set of quandle representations of the knot quandle \(Q_K\) into the Alexander quandle \({\mathrm{AQ}_{t}}\). These ideas extend naturally from knots to tangles, leading to a Topological Quantum Field Theory, developed in [10], that encodes the local and global properties of the Alexander module in terms of representations:

We give a brief overview of the main ideas of this construction in Section 2, but let us emphasize that the target category of Theorem 2 makes possible to apply linear algebra (within the category of spans of complex vector spaces) to compute Alexander invariants of tangles, a feature already present in the subcategory of braids through the Burau representation.

This manuscript applies Theorems 1 and 2 to two classical and well-studied families in knot theory, namely rational knots and pretzel knots (and their associated tangles), in analogy with other TQFTs [14]. As these families are well studied, our approach recovers several classical results from a different perspective and also produces new formulas and insights. First, we study the \({\mathrm{AGL}_{1}}(\mathbb{C})\)-representations of elementary oriented 2-tangles and their compositions, which give rise to codirected and alternating braidings, and we express them in terms of quantum integers, \([n]_t=\frac{1-t^n}{1-t}\). A careful analysis of the way these tangles are glued to form rational and pretzel knots and tangles leads us to closed-form expressions for their Alexander modules. Focusing on their Alexander polynomials, we obtain:

For pretzel knots, we derive the following formulas in the even and odd cases:

Closed-form expressions for the Alexander polynomial of these families have previously appeared in the literature. For instance, a continuant expression for the Conway polynomial can be obtained from the skein relation [15], and closed formulas for the polynomials of general pretzel knots have recently been derived in the same manner [16], coinciding with our results in the odd case. Particular cases have also been studied (see [17], [18]), and a connection with Fibonacci polynomials in this context was established in [19]. In contrast, our method, which arises from our TQFT framework, determines the full Alexander module, thus revealing additional structural features. For example, the pretzel formulas are naturally deduced from a continuant recurrence that reflects the topological construction of pretzel tangles, leading via the functor \({\mathcal{A}}_t\) to a cyclic tridiagonal matrix that matches the cyclic nature of pretzel knots.

Using these formulas, we are able to analyze the zeros of the Alexander polynomial in these families. We recall the following conjecture:

The conjecture was proved by Ishikawa to be true for rational knots [20], and false for general knots [21], [22], after intense research regarding the location of the zeros of the Alexander polynomial [23]–[25]. Our results provide alternative proofs of this conjecture for pretzel knots that admit an alternating diagram, providing stronger bounds on the zeros of the Alexander polynomial in some of these pretzel families. Specifically, we prove the following:

Finally, we show that the specialization of the functor \({\mathcal{A}}_t\) at \(t=-1\) recovers a classical result in knot theory, namely the classification of rational tangles. This demonstrates that the TQFT developed in [10] complements existing approaches in the literature, such as Conway’s original method via the Alexander-Conway polynomial [26], or the topological and arithmetic proofs via colorings due to Kauffman and Lambropoulou [27]. Our perspective emphasizes how the span category, together with the rotation of spans induced by the rotation of 2-tangles, provides a natural framework to “linearize’’ tangles which are not braids, taking into account the image of the embedding \(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}} \hookrightarrow \mathop{\mathrm{Span}}(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}})\). Concretely, we establish the following:

This leads to a reinterpretation of Kauffman–Lambropoulou’s proof of the classification of rational tangles via colorings in terms of spans and the geometry of the complex Grassmannian \(\mathop{\mathrm{Gr}}(2,4)\). In particular, we prove:

Structure of the manuscript↩︎

Section 2 provides a brief overview of the construction of the functors \({\mathcal{A}}\) and \({\mathcal{A}}_t\), together with explicit generating sets for the relevant groups and categories, which will be needed for computations in later sections. Section 3 analyzes the action of \({\mathcal{A}}_t\) on elementary tangles, with emphasis on alternating and codirected braidings, which serve as the building blocks in the decompositions of 2-bridge knots and pretzel knots. These two families are treated in Sections 4 and 5, where explicit formulas for their Alexander polynomials are derived and their zeros are analyzed. Finally, Section 6 is devoted to the specialization \({\mathcal{A}}_{-1}\) and its connection with rational tangles.

Acknowledgements↩︎

The author thanks Ángel González Prieto and Vicente Muñoz for many valuable discussions on this work, and Joan Porti and Pedro M. González Manchón for suggestions and conversations regarding pretzel knots.

2 \({\mathrm{AGL}_{1}}(\mathbb{C})\)-representations and Alexander invariants↩︎

In this section we review the construction of the functors \({\mathcal{A}}_t\) and \({\mathcal{A}}\) and state some of their properties that are used throughout the paper. We begin by recalling the relevant categories, then outline the construction of these functors and, after fixing generators, carry out explicit computations. For full details, see [10].

The tangle category↩︎

Let \({\mathcal{T}}\) be the category of tangles, whose objects are sequences of \(\pm\) signs, possibly empty, which we geometrically realize as a set of evenly spaced points in \(\mathbb{R}^2\) with integer coordinates. An oriented tangle \(T\) is defined as the union any finite number of disjoint polygonal arcs in \(\mathbb{R}^2\times [0,1]\) such that the boundary satisfies the condition \[\partial T=T \cap\left(\mathbb{R}^2 \times\{0,1\}\right)=([k] \times\{0\} \times\{0\}) \cup([\ell] \times\{0\} \times\{1\}),\] where \([n]=\lbrace 1, \ldots, n\rbrace\) and \([0]=\emptyset\). The isotopy classes of these oriented tangles define the morphisms of the category \(\mathcal{T}\), between the source of \(T\), that is, \(\partial T\cap (\mathbb{R}^2\times \lbrace 0 \rbrace)\), and the target of \(T\), \(\partial T\cap (\mathbb{R}^2\times \lbrace 1 \rbrace)\). The composition of two tangles \(T\) and \(T'\), when defined, is the isotopy class of the tangle that is obtained by placing \(T'\) on top of \(T\), properly rescaled to produce a tangle in \(\mathbb{R}^2\times [0,1]\). Concatenation of tangles endows the category \({\mathcal{T}}\) with a non-symmetric braided monoidal structure. We recall that the category \({\mathcal{T}}\) can be presented in terms of generators and relations [28]. Explicitly, it is generated by the six elementary morphisms \[\label{eqn:generatorstanglecat} \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \draw[thick,->] (0,1) .. controls (0,-0.2) and (1,-0.2) .. (1,1); \end{tikzpicture}, \, \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \draw[thick,<-] (0,1) .. controls (0,-0.2) and (1,-0.2) .. (1,1); \end{tikzpicture},\, \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \coordinate (base) at (0,0); \draw[thick,->] (0,0.1) .. controls (0.1,1.3) and (1,1.3) .. (1,0.1); \end{tikzpicture} , \, \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \coordinate (base) at (0,0); \draw[thick,<-] (0,0.1) .. controls (0.1,1.3) and (1,1.3) .. (1,0.1); \end{tikzpicture} , \, X_{+}, \, X_{-},\tag{2}\] subject to relations that arise from Redemeister transformations and isotopies of tangle diagrams.

Quandles↩︎

A quandle is a set \(Q\) equipped with a binary operation \(\triangleright: Q\times Q \longrightarrow Q\) satisfying:

• \(x\triangleright x= x\), for all \(x\in Q\).

• For all \(x,z\in Q\), there exists a unique \(y\in Q\) such that \(x\triangleright y=z\).

• \(x\triangleright (y\triangleright z) = (x\triangleright y)\triangleright (x\triangleright z)\), for all \(x,y,z\in Q\).

A quandle homomorphism is a map \(f:Q\rightarrow Q'\) preserving the quandle operation, i.e.\(f(x\triangleright_Q y)=f(x)\triangleright_{Q'}f(y)\) for all \(x,y \in Q\), so that quandles define a category, denoted by \(\boldsymbol{\mathrm{Qdl}}\). We note that uniqueness in (P2) leads to a second operation \(\triangleleft\), defined as \(z \triangleleft x = y\) if and only if \(x \triangleright y =z\).

We note that a quandle can be constructed from any group \(G\) by defining the binary operation \[a \triangleright b = aba^{-1}, \quad \text{for } a,b \in G,\] which endows \(G\) with the structure of a quandle, known as the conjugation quandle, and denoted \(\mathop{\mathrm{Conj}}(G)\). Conversely, for any quandle \(Q\), one defines the associated group as \[\mathop{\mathrm{As}}(Q) = \left\langle a \in Q \,\middle|\, a \triangleright b = aba^{-1},\;\text{for all } a,b \in Q \right\rangle,\] which encodes the quandle operation in group-theoretic terms.

These constructions define an adjunction between the category of groups \(\mathbf{Grp}\) and the category of quandles \(\boldsymbol{\mathrm{Qdl}}\) \[\label{eqn:adjunctiongrqdl} \mathop{\mathrm{Hom\,}}_{\mathbf{Grp}}(\mathop{\mathrm{As}}(Q), G) \cong \mathop{\mathrm{Hom\,}}_{\boldsymbol{\mathrm{Qdl}}}(Q, \mathop{\mathrm{Conj}}(G)),\tag{3}\] for any group \(G\) and quandle \(Q\) [29]. We are interested in the following quandles:

• The knot quandle \(Q_K\) of any knot \(K\), introduced by Joyce and Matveev [30], [31]. It is a complete knot invariant up to orientation, and it can be described geometrically or algebraically, in terms of a planar projection of the knot, providing a Wirtinger-type presentation of the quandle in terms of generators and relations, so that \(\mathop{\mathrm{As}}(Q_K)\cong \pi_1(S^3-K)\).

• Given any set \(S\) and the free group \(\mathbf{F}_{S}\) that it generates, the free quandle on \(S\) is the set \(\boldsymbol{FQ}_S \subset \mathbf{F}_{S}\) consisting of all conjugates in \(\mathbf{F}_{S}\) of the elements of \(S\), that is, \[\boldsymbol{FQ}_S=\{ w s w^{-1} | w\in \mathbf{F}_{S}, s\in S\}.\] The quandle operation is defined by \(a \triangleright b= aba^{-1}\), so that the free quandle is a subquandle of \(\mathop{\mathrm{Conj}}(\mathbf{F}_S)\).

• Given any \(t \in {\mathbb{C}}^* = {\mathbb{C}}- \{0\}\), the \(t\)-Alexander quandle is \({\mathrm{AQ}_{t}} = ({\mathbb{C}}, \triangleright_t)\) with the operations \[\label{eqn:Alexquandlerel} a\triangleright_t b=tb+(1-t)a, \qquad b \triangleleft_t a = t^{-1}b + (1-t^{-1})a,\tag{4}\] for \(a, b \in {\mathbb{C}}\). This Alexander quandle is subquandle of \(\mathop{\mathrm{Conj}}({\mathrm{AGL}_{1}})\), via the quandle morphism \[{\mathrm{AQ}_{t}} \to \mathop{\mathrm{Conj}}({\mathrm{AGL}_{1}}), \qquad a \mapsto \begin{pmatrix} 1 & 0 \\ a & t \end{pmatrix}.\]

The Category \(\mathop{\mathrm{Span}}(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}})\)↩︎

We briefly recall the main features of the category \(\mathop{\mathrm{Span}}(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}})\), where \(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}}\) denotes the category of finite-dimensional complex vector spaces:

• Objects: same as in \(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}}\); that is, finite-dimensional complex vector spaces.

• Morphisms: For \(X, Y \in \mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}}\), a morphism from \(X\) to \(Y\) is represented by a span \((Z,f,g)\) of linear maps, \[\xymatrix{ & Z \ar[dl]_{f} \ar[dr]^{g} & \\ X & & Y }\] i.e., by linear maps \(f: Z \to X\), \(g: Z \to Y\). Morphisms are equivalence classes of spans: \((Z, f, g) \sim (Z', f', g')\) if there exists an isomorphism \(\varphi: Z \to Z'\) such that \(f = f' \circ \varphi\) and \(g = g' \circ \varphi\).

• Composition: Given morphisms \((Z_1, f_1, g_1): X \to Y\) and \((Z_2, f_2, g_2): Y \to Z\), their composition is defined by the pullback span \[\xymatrix{ & & Z_1 \times_Y Z_2 \ar[dl]_{\pi_1} \ar[dr]^{\pi_2} & & \\ & Z_1 \ar[dl]_{f_1} \ar[dr]^{g_1} & & Z_2 \ar[dl]_{f_2} \ar[dr]^{g_2} & \\ X & & Y & & Z }\] where \(Z_1 \times_Y Z_2 = \{(z_1, z_2) \in Z_1 \times Z_2 \mid g_1(z_1) = f_2(z_2)\}\), the fibre product in \(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}}\).

Finally, we point out that the category \(\mathop{\mathrm{\mathbf{Vect}}}_{\mathbb{C}}\) is a monoidal category with the direct sum \(\oplus\) of vector spaces. This monoidal structure is inherited by \(\mathop{\mathrm{Span}}(\mathop{\mathrm{\mathbf{Vect}}}_{\mathbb{C}})\) through direct sum of both objects and spans.

The functors \({\mathcal{A}}\) and \({\mathcal{A}}_t\)↩︎

We outline now the main ideas behind the construction of the functors \({\mathcal{A}}\), \({\mathcal{A}}_t\), and we choose generators for our computations, following [10].

Given any tangle \(T\), we can associate to it the span of \({\mathrm{AGL}_{1}}(\mathbb{C})\)-representation varieties \[\label{eqn:spanAGL1repvarieties} \xymatrix{ & {\mathcal{R}}_{{\mathrm{AGL}_{1}}(\mathbb{C})}(\Gamma_T) \ar[dl]_{i} \ar[dr]^{j} & \\ {\mathcal{R}}_{{\mathrm{AGL}_{1}}(\mathbb{C})}(\Gamma_\epsilon) & & {\mathcal{R}}_{{\mathrm{AGL}_{1}}(\mathbb{C})}(\Gamma_\varepsilon) }\tag{5}\] where \(\epsilon\) and \(\varepsilon\) are the boundary points \(\partial T=\overline{\epsilon} \sqcup \varepsilon\), and \(\Gamma_T\), \(\Gamma_\epsilon\) and \(\Gamma_{\varepsilon}\) are fundamental groups of the complement of the tangle in \({\mathbb{R}}^2\times [0,1]\) and its boundary points in \(\mathbb{R}^2\). The maps \(i: {\mathcal{R}}_{{\mathrm{AGL}_{1}}(\mathbb{C})}(\Gamma_T) \to {\mathcal{R}}_{{\mathrm{AGL}_{1}}(\mathbb{C})}(\Gamma_\epsilon)\) and \(j: {\mathcal{R}}_{{\mathrm{AGL}_{1}}(\mathbb{C})}(\Gamma_T) \to {\mathcal{R}}_{{\mathrm{AGL}_{1}}(\mathbb{C})}(\Gamma_\varepsilon)\) are maps at the level of representation spaces which are naturally induced by the inclusion of the boundaries in \(T\). Besides, each of these representation varieties can be regarded as singular vector bundle, as there always is a regular morphism between representation varieties \[\label{eq:Alexander-fibration-intro} \pi: {\mathcal{R}}_{{\mathrm{AGL}_{1}}}(\Gamma) \longrightarrow {\mathcal{R}}_{{\mathbb{C}}^{\ast}}(\Gamma) = {\mathbb{C}}^*,\tag{6}\] the Alexander fibration, induced by the projection of \(\pi: {\mathrm{AGL}_{1}}(\mathbb{C})\rightarrow {\mathbb{C}}^*\) onto its diagonal entry. The maps \(i\) and \(j\) are compatible with their fibered structures, and the Alexander polynomial arises as the generator of the vanishing ideal of the exceptional locus of this fibration.

Fix any \(t \in {\mathcal{R}}_{{\mathbb{C}}^{\ast}}(\Gamma_K) = {\mathbb{C}}^*\). Considering the fibre \(\pi^{-1}_K(t)\) naturally leads to the functor \({\mathcal{A}}_t\) via its connection to quandle theory. When \(t\in {\mathbb{C}}^{\ast}\) is fixed in 5 , the adjunction 3 yields a span of quandle representation varieties \[\label{eqn:spanAQquandles} \xymatrix{ & {\mathcal{R}}_{{\mathrm{AQ}_{t}}}(Q_T) \ar[dl]_{i} \ar[dr]^{j} & \\ {\mathcal{R}}_{{\mathrm{AQ}_{t}}}(Q_\epsilon) & & {\mathcal{R}}_{{\mathrm{AQ}_{t}}}(Q_\varepsilon) }\tag{7}\] where \(Q_T, Q_{\epsilon}\) and \(Q_{\varepsilon}\) are the quandles associated to each of these spaces. In other words, the inclusion of the boundaries of \(T\) produces a span of quandles, which we represent onto the Alexander quandle \({\mathrm{AQ}_{t}}\) [32]. In particular, when \(K\) is a knot, \(\pi_K^{-1}(t)\) can be identified with the quandle representation variety \[{\mathcal{R}}_{{\mathrm{AQ}_{t}}}(Q_K) = \mathop{\mathrm{Hom\,}}_\boldsymbol{\mathrm{Qdl}}(Q_K, {\mathrm{AQ}_{t}}),\] of the knot quandle.

\({\mathcal{A}}_t\) on the set of generators of \({\mathcal{T}}\)↩︎

We describe now explicitly the functor \({\mathcal{A}}_t\) vía its action on the generators of \({\mathcal{T}}\) listed in 2 , using a natural presentation of the associated fundamental groups as follows.

• Objects. Fix an object \(\epsilon\in\mathcal{T}\) with length equal to \(n:=\abs{\epsilon}\). The complement \(\epsilon^{c}\) in \(\mathbb{R}^2\) has fundamental group \(\Gamma_\epsilon:=\pi_1(\epsilon^{c})\cong \mathbf{F}_{n}\). We choose orientations of the standard generators according to the sign of the corresponding point, so that from a top view the loops wind clockwise or counterclockwise (see Figure 1).

  

  The same criterion can be chosen for the quandle \(Q_\epsilon\) when considering representatives of the set of homotopy classes of paths from a fixed basepoint to a neighborhood of \(\epsilon\), so that we get a natural identification \({\mathcal{A}}_t(\epsilon) = {\mathcal{R}}_{{\mathrm{AQ}_{t}}}(\epsilon^c) = {\mathbb{C}}^n\).

• Morphisms. Fix a tangle \(T:\epsilon\to\varepsilon\). The complement \(T^{c}\) has fundamental group \(\Gamma_T:=\pi_1(T^{c})\). We take as generators the meridians encircling each strand, oriented by the right–hand rule (see Figure 1). Again, this also defines generators of the quandle \(Q_T\), leading to a description of \({\mathcal{R}}_{{\mathrm{AQ}_{t}}}(T)\).

Generators↩︎

With these choices on the fundamental group and the associated quandle for the six generators of \({\mathcal{T}}\), we obtain the oriented generators depicted in Figure 2.

It is proved in [10] that the functor \({\mathcal{A}}_t\) on these tangles produces the following spans of complex vector spaces:

• Braids \(X_+,X_-\): \({\mathcal{A}}_t(X_+)\) and \({\mathcal{A}}_t(X_-)\) are respectively given by the spans \[\xymatrix{ & {\mathbb{C}}^2 \ar[ld]_{\mathop{\mathrm{Id}}} \ar[rd]^{f_+}& \\ {\mathbb{C}}^2 & & {\mathbb{C}}^2 } \quad \xymatrix{ & {\mathbb{C}}^2 \ar[ld]_{\mathop{\mathrm{Id}}} \ar[rd]^{f_-}& \\ {\mathbb{C}}^2 & & {\mathbb{C}}^2 }\] where \(f_+,f_-:{\mathbb{C}}^2\longrightarrow {\mathbb{C}}^2\) are defined as: \[\begin{align} f_+(a_1, a_2) & =((1-t)a_1 + ta_2, a_1) \tag{8}\\ f_-(a_1, a_2) & =(a_2, t^{-1}a_1+(1-t^{-1})a_2) \tag{9} \end{align}\]

• Evaluations and coevaluations: it is shown that \({\mathcal{A}}_t( \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \coordinate (base) at (0,0); \draw[thick,<-] (0,0.1) .. controls (0.1,1.3) and (1,1.3) .. (1,0.1); \end{tikzpicture} ) = {\mathcal{A}}_t( \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \coordinate (base) at (0,0); \draw[thick,->] (0,0.1) .. controls (0.1,1.3) and (1,1.3) .. (1,0.1); \end{tikzpicture} )\) and also \({\mathcal{A}}_t( \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \draw[thick,<-] (0,1) .. controls (0,-0.2) and (1,-0.2) .. (1,1); \end{tikzpicture})={\mathcal{A}}_t( \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \draw[thick,->] (0,1) .. controls (0,-0.2) and (1,-0.2) .. (1,1); \end{tikzpicture})\) are respectively given by the spans \[\xymatrix{ & {\mathbb{C}}\ar[ld]_{\Delta} \ar[rd]& \\ {\mathbb{C}}^2 & & 0 }\quad \xymatrix{ & {\mathbb{C}}\ar[rd]^{\Delta} \ar[ld]& \\ 0 & & {\mathbb{C}}^2 }\] where \(\Delta:{\mathbb{C}}\longrightarrow {\mathbb{C}}^2\) is the diagonal map \(\Delta(a)=(a,a)\).

  We note that this last case naturally generalizes to the span attached to the evaluation and coevaluation maps in \({\mathcal{T}}\). That is, given any object \(\epsilon=(\epsilon_1,\ldots,\epsilon_n) \in {\mathcal{T}}\) of length \(n\), there is an evaluation tangle \(\mathop{\mathrm{ev}}:\epsilon \, \otimes \epsilon^{\ast} \rightarrow \emptyset\), where \(\epsilon^{\ast}=(-\epsilon_n,\ldots,-\epsilon_1)\) is the dual object. A similar construction produces the coevaluation tangle \(\mathop{\mathrm{coev}}:\emptyset \rightarrow\epsilon \, \otimes \epsilon^{\ast}\) The spans \({\mathcal{A}}_t(\mathop{\mathrm{ev}}_{\epsilon})\) and \({\mathcal{A}}_t(\mathop{\mathrm{coev}}_{\epsilon})\) are respectively equal to \[\label{eqn:evandcoevspans} \xymatrix{ & {\mathbb{C}}^n \ar[ld]_{(\mathop{\mathrm{Id}},c)} \ar[rd]& \\ {\mathbb{C}}^{2n} & & 0 } \quad \xymatrix{ & {\mathbb{C}}^n \ar[ld] \ar[rd]^{(\mathop{\mathrm{Id}},c)}& \\ 0 & & {\mathbb{C}}^{2n} }\tag{10}\] where \(c(z_1,\ldots,z_n)=(z_n,\ldots,z_1)\).

  The transpose \(T^{\ast}\) of a tangle \(T\), seen as a morphism between \(\epsilon\) and \(\varepsilon\), is the tangle \(T^{\ast}=(\mathop{\mathrm{coev}}_{\epsilon}\otimes \mathop{\mathrm{Id}}_{\varepsilon}) \circ (\mathop{\mathrm{Id}}_{\epsilon}\otimes \, T\otimes \mathop{\mathrm{Id}}_{\varepsilon}) \circ (\mathop{\mathrm{Id}}_{\epsilon}\otimes \mathop{\mathrm{ev}}_{\varepsilon})\), where \(\mathop{\mathrm{Id}}_{\epsilon}=(\mathop{\mathrm{Id}})^{\otimes \abs{\epsilon}}\) is properly oriented. If \({\mathcal{A}}_t(T)\) is the span \({\mathbb{C}}^r \overset{f}{\longleftarrow} {\mathbb{C}}^n \overset{g}{\longrightarrow} {\mathbb{C}}^s\), then Equation 10 implies that \({\mathcal{A}}_t(T^{\ast})\) is the following span \[\label{eqn:transposespan} \xymatrix{ & {\mathbb{C}}^n \ar[ld]_{c \, \circ \, g} \ar[rd]^{c \, \circ \, f}& \\ {\mathbb{C}}^s & & {\mathbb{C}}^{r} }\tag{11}\]

2.1 Quantum integers↩︎

Finally, we introduce notation and basic facts regarding the quantum integers \([n]_{t}\) that will be used throughout the text. Quantum integers are Laurent polynomials in \({\mathbb{Z}}[t,t^{-1}]\) that are defined as \[[n]_{t}:=\frac{1-t^n}{1-t}=t^{n-1}+\ldots+1,\] where \(n\in {\mathbb{Z}}\) and we set \([0]_t=0\). We will make use of the following properties, which will be mostly used evaluated at \(-t\):

• (Recursiveness) \[\label{property:quantumint1} t[n]_t+1=[n+1]_{t}, \quad [1]_t=1.\tag{12}\]

• (Negative integers) \[\label{property:quantumint2} [-n]_t=-\frac{1}{t}-\frac{1}{t^2}-\ldots-\frac{1}{t^n}=-t^{-n}[n]_t\tag{13}\]

• (Evaluation at \(t^{-1}\)) \[\label{property:quantumint3} [n]_{t^{-1}}=t^{1-n}[n]_t=-t[-n]_t\tag{14}\]

ff

3 2-tangles↩︎

We discuss in this section the effect of the functor \(\mathcal{A}_t\) on 2-tangles, starting with the generators \(X_+\) and \(X_-\) of the tangle category and the braidings they generate. The analysis of these elementary tangles will allow us to study rational and pretzel tangles and links in Sections 4 and 5.

Rotation and addition↩︎

Applying a counterclockwise rotation by an angle of \(\pi/2\) to \(X_+\) and \(X_-\), introduced in Section ¿sec:sec:generatorsdescription?, yields eight elementary tangles \[\begin{tikzpicture}[scale=0.4,baseline=0.5ex] \coordinate (base) at (0,-0.5); \draw[thick,->] (1,0) -- (0,1); \draw[white,line width=4pt] (0,0) -- (1,1); \draw[thick,->] (0,0) -- (1,1); \end{tikzpicture} , \begin{tikzpicture}[scale=0.4,baseline=0.5ex] \coordinate (base) at (0,-0.5); \draw[thick,->] (1,1) -- (0,0); \draw[white,line width=4pt] (0,1) -- (1,0); \draw[thick,->] (1,0) -- (0,1); \end{tikzpicture} , \begin{tikzpicture}[scale=0.4,baseline=0.5ex] \coordinate (base) at (0,-0.5); \draw[thick,->] (0,1) -- (1,0); \draw[white,line width=4pt] (0,0) -- (1,1); \draw[thick,->] (1,1) -- (0,0); \end{tikzpicture} , \begin{tikzpicture}[scale=0.4,baseline=0.5ex] \coordinate (base) at (0,-0.5); \draw[thick,->] (0,0) -- (1,1); \draw[white,line width=4pt] (0,1) -- (1,0); \draw[thick,->] (0,1) -- (1,0); \end{tikzpicture} , \begin{tikzpicture}[scale=0.4,baseline=0.5ex] \coordinate (base) at (0,-0.5); \draw[thick,->] (0,0) -- (1,1); \draw[white,line width=4pt] (0,1) -- (1,0); \draw[thick,->] (1,0) -- (0,1); \end{tikzpicture} , \begin{tikzpicture}[scale=0.4,baseline=0.5ex] \coordinate (base) at (0,-0.5); \draw[thick,->] (1,0) -- (0,1); \draw[white,line width=4pt] (0,0) -- (1,1); \draw[thick,->] (1,1) -- (0,0); \end{tikzpicture} , \begin{tikzpicture}[scale=0.4,baseline=0.5ex] \coordinate (base) at (0,-0.5); \draw[thick,->] (1,1) -- (0,0); \draw[white,line width=4pt] (0,1) -- (1,0); \draw[thick,->] (0,1) -- (1,0); \end{tikzpicture} , \begin{tikzpicture}[scale=0.4,baseline=0.5ex] \coordinate (base) at (0,-0.5); \draw[thick,->] (0,1) -- (1,0); \draw[white,line width=4pt] (0,0) -- (1,1); \draw[thick,->] (0,0) -- (1,1); \end{tikzpicture}\] which differ from \(X_+\) and \(X_-\) only in their orientations. Together with the remaining generators \(\begin{tikzpicture}[scale=0.35,baseline=0.5ex] \draw[thick,->] (0,1) .. controls (0,-0.2) and (1,-0.2) .. (1,1); \end{tikzpicture}, \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \draw[thick,<-] (0,1) .. controls (0,-0.2) and (1,-0.2) .. (1,1); \end{tikzpicture}, \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \coordinate (base) at (0,0); \draw[thick,->] (0,0.1) .. controls (0.1,1.3) and (1,1.3) .. (1,0.1); \end{tikzpicture} , \begin{tikzpicture}[scale=0.35,baseline=0.5ex] \coordinate (base) at (0,0); \draw[thick,<-] (0,0.1) .. controls (0.1,1.3) and (1,1.3) .. (1,0.1); \end{tikzpicture}\), this collection is sometimes considered as a generating set for the category of tangle diagrams; see [28].

The examples above are 2-tangles: tangles with two interlacing strands and no loops, so that \(\abs{b(T)}=\abs{s(T)}=2\). We index the endpoints by NE, NW, SE and SW according to their position in a planar diagram. The category of 2-tangles \({\mathcal{T}}_2\) is special because it can be endowed with additional operations:

• The sum \(T+T'\) of two tangles, which is the new tangle that is diagrammatically formed by attaching the NW and SW strands of \(T\) to the NE and SE strands of \(T'\), see Figure 3. It is defined when \(SW(T)=-SE(T')\) and \(NW(T)=-NE(T')\).

• The rotation \(T^r\) of a tangle \(T\), which is the tangle that arises by performing a counter-clockwise rotation of angle \(\pi/2\) on the plane.

• The mirror image \(\overline{T}\), obtained by switching all the crossings in \(T\).

In terms of a tangle decomposition, we can also define the rotation of a 2-tangle \(T\) as follows.

The sum of two tangles \(T,T'\) can be defined in terms of the rotation map as follows.

The following lemma, which we state without proof, summarizes the effect of rotating a 2-tangle \(T\) at the level of spans. The result can be obtained by directly applying the functor \({\mathcal{A}}_t\) to the tangle \(R(T)\):

The rotation \(R\) has order four and restricts to an involution on unoriented rational tangles [27]. Moreover, for \(2\)-tangles we have \(R^{2}(T)=T^{\ast}\), so the span \(\mathcal{A}_t(R^{2}(T))\) can be described via Lemma 5 or Equation 11 . Analyzing the action of \(R\) on spans through the functor \(\mathcal{A}_t\) we obtain the following proposition.

We may observe that each leg of the spans that appear in Proposition 8 is an isomorphism. Therefore, we can restate Proposition 8 in terms of basic spans, providing an second description of the effect of rotating a span with respect to the functor \({\mathcal{A}}_t\). To ease notation, given linear maps \(f,g:{\mathbb{C}}^2\longrightarrow {\mathbb{C}}^2\), let \((f,g)\) denote the span \({\mathbb{C}}^2 \overset{f}{\longleftarrow} {\mathbb{C}}^2 \overset{g}{\longrightarrow} {\mathbb{C}}^2\). Besides, let \(C(f)\) be the linear map \(c\circ f \circ c\) and let \(I(f)=f^{-1}\).

Extending the usual notion for knots, we define the reverse of a tangle as the new tangle that originates when the orientation of every strand is reversed. The following corollary, expected from the nature of the Alexander module, also stems from Proposition 9:

Braidings↩︎

We analyze in this subsection the image under \({\mathcal{A}}_t\) of iterated compositions of the elementary tangles \(X_+, X_{-}\) and their rotations. These oriented twisted braidings, formed by two interlacing strands, are shown in Figure 4, and they are the basic building blocks in which we may decompose any rational tangle, presented in its 2-bridge form, or any pretzel tangle.

Since orientations play an important role in the functor \({\mathcal{A}}_t\), we distinguish two types of oriented twisted braidings: codirected (which are oriented braids) and alternating (strands have opposite orientation). We recall that \(f_{+}(t)\) and \(f_-(t)\), defined in 15 , are the linear maps of the basic spans \({\mathcal{A}}_t(X_+)\) and \({\mathcal{A}}_t(X_-)\), we emphasize here their dependence on \(t\in {\mathbb{C}}^{\ast}\).

We will use two equivalent expressions for the maps \(f^n_+\) and \(f^n_-\), specialized either at \(t^{-1}\) or with \(t\)-factors omitted, as stated in the following corollary:

Alternating positive braidings are compositions of the tangles \(R(X_+)\) and \(R^3(X_+)\). Their image by \({\mathcal{A}}_t\)was computed in 8 and is respectively given by the spans \[\label{eqn:RX43R3X43spans} \xymatrix{ & {\mathbb{C}}^2 \ar[ld]_{f_+(t)} \ar[rd]^{\mathop{\mathrm{Id}}}& \\ {\mathbb{C}}^2 & & {\mathbb{C}}^2 }\qquad \xymatrix{ & {\mathbb{C}}^2 \ar[ld]_{c} \ar[rd]^{c\; \circ f_{+}(t)}& \\ {\mathbb{C}}^2 & & {\mathbb{C}}^2 }\tag{16}\]

We may also define the negatively oriented version of \(\widetilde{T}^{n}_+\), that is:

• \(\widetilde{T}_-^{2k}=(R^3(X_-)\circ R(X_-))^k\).

• \(\widetilde{T}_-^{2k+1}=R(X_-)\circ (R^3(X_-)\circ R(X_-))^k\).

The following proposition computes their associated basic spans:

4 2-bridge knots↩︎

Starting from the unoriented trivial \(2\)-tangle, one can construct new tangles—called rational tangles—by composing and adding powers of the basic tangles \(X_+\) and \(X_-\) and their rotations. Equivalently, a rational tangle is any tangle produced by successively applying a finite sequence of horizontal and vertical twists \(a_1,\ldots,a_n\) to the four endpoints of the trivial tangle or its rotation (\([\infty]\) and \([0]\), respectively).

These tangles were classified by Conway [26], who established a bijection between them and the set of extended rational numbers \(\mathbb{Q}\cup\{\infty\}\) via continued fraction expansions: \[\frac{p}{q} = [a_1,\ldots,a_n] = a_1 + \cfrac{1}{a_{2}+\cfrac{1}{\ddots + \cfrac{1}{a_n}}} \;\longmapsto\; T_{[a_1,\ldots,a_n]}.\] Conway observed that the arithmetic of these tangles, under addition, inversion, and composition, mirrors the arithmetic of rational numbers. We will revisit them in Section 6 from their coloring perspective [27].

Closing adjacent endpoints of any rational tangle (applying a numerator or denominator closure) produces a rational knot, or a 2-bridge knot \(b(p,q)\). These notions are equivalent, and the resulting links were first classified by Schubert using 2-fold branched covers. In what follows, we will use 2-bridge presentations of a rational knot \(K\). Starting from a 3-strand braid model of a rational tangle \(T=[a_1,\ldots,a_n]\), we tensor with a trivial strand and obtain the rational knot \(b(p,q)\) by taking the appropiate closure determined by the parity of \(n\), see Figure 5:

Propositions 14, 16 and 18 will enable us to compute and study the Alexander invariants of any rational knot, via the functor \({\mathcal{A}}_t\). We begin by recalling the notion of the equalizer of a set of linear maps.

When \(\mathcal{F}=\lbrace f,g\rbrace\) consists of two maps, we write \(\mathop{\mathrm{Eq}}(f,g)\); this is also known in the literature as the difference kernel. The next lemma details the effect of closing a rational tangle in the \(\mathop{\mathrm{Span}}(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}})\) category.

Propositions 14, 16, 18 and Remark 21 provide the necessary ingredients to directly compute \({\mathcal{A}}_t(K)\) from an oriented planar projection of \(K\), as the next example shows.

Example 22 shows that \({\mathcal{A}}_t\) computes the Alexander module of a knot \(K\) from any planar diagram. If \(K\) is a rational knot with associated fraction \(p/q\), a continued-fraction expression of the form \[\frac{p}{q}=[a_1,\ldots,a_n], \quad a_i\in \mathbb{Z}_{+}, \quad n \text{ odd},\] is said to be in canonical form; this representation is unique [27]. Other generalized Euclidean algorithms yield different, but again unique, continued-fraction expansions obtained by imposing certain restrictions on \(a_i\). For our purposes, we recall the following expansion, which is particularly convenient for rational tangles [15]:

Let \(K\) be a rational knot presented as the numerator closure of a rational tangle \(T_{p/q}\). Any continued-fraction expansion of \(p/q\) gives rise to a planar 2-bridge diagram for \(K\). If the expansion has even length and all coefficients are even, then every braiding block in the diagram is alternating. We may orient \(K\) so that, in the tangle corresponding to the terminal coefficient \(2k_{1}\), the first generator is \(R^3(X_\pm)\). With this choice, the first twist in each subsequent box associated to \(2k_i\) alternates between \(R^3(X_\pm)\) and \(R(X_\pm)\) as \(i\) increases. As is customary for planar diagrams of rational knots, when \(2k_i>0\), (unoriented) twists are positive for even \(i\) and negative for odd \(i\); the signs are reversed when \(2k_i<0\). These conventions lead to the following proposition:

Proposition 25 may be used to obtain a recursive formula for the Alexander polynomial of any rational knot from the description of \({\mathcal{A}}_t(K)\). We consider the even continued-fraction expansion \(p/q=[2k_1,\ldots,2k_{2n}]\) under the previous assumptions, and we use the standard basis for the 3-strand braid model of the rational tangle \(T_{p/q}\), following the conventions set in Section 2. Let us write \[A_{2i}=\mathop{\mathrm{diag}}(1,A'_{2i},1), \qquad A_{2i-1}=\mathop{\mathrm{diag}}(1,1,A'_{2i-1}), \quad H=\begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix},\] where \[\begin{align} & A'_{2i-1} =\begin{pmatrix} 1+k_{2i-1}(1-t) & -k_{2i-1}(1-t) \\ k_{2i-1}(1-t) & 1-k_{2i-1}(1-t) \end{pmatrix}=\mathop{\mathrm{Id}}+k'_{2i-1}H, \qquad &k'_{2i-1} :=k_{2i-1}(1-t), \\ & A'_{2i} =\begin{pmatrix}1-k_{2i}(1-t^{-1}) & k_{2i}(1-t^{-1})\\ -k_{2i}(1-t^{-1}) & 1+k_{2i}(1-t^{-1}) \end{pmatrix}=\mathop{\mathrm{Id}}+k'_{2i}H, \qquad & k'_{2i} :=-k_{2i}(1-t^{-1}), \end{align}\]

where we also recall that \(h\) is the nilpotent map defined in Proposition 16. Let us also write \(P,Q\) for the matrices from 17 that correspond to the closure of the rational tangle T. Proposition 25 establishes that \({\mathcal{A}}_t(K)\) is given by the span \[\xymatrix{ & \ker N \ar[ld] \ar[rd]& \\ 0 & & 0 }\] where \[\label{eqn:PA95iQ} N=PA_{2n}A_{2n-1}\ldots A_{2}A_{1}Q,\tag{18}\] We shall obtain an explicit description in terms of continuants of the Alexander polynomial of \(T\). We recall their definition:

In this section, we will restrict to the case where \(u_i=l_i=1\) for all \(i\), first studied by Euler and classically connected with continued fractions; we will write \(K_n:=K_n(a_1,\ldots,a_n)\).

In terms of quantum integers, we may restate the theorem as follows.

5 Pretzel knots↩︎

Pretzel knots are a well-studied family of knots where Propositions 14, 16, 18 may be directly applied to determine their Alexander modules. A Pretzel link \(P(q_1,\ldots,q_n)\) is a link that is obtained by cyclically connecting \(n\) twisted bands, where each one has \(q_i\) half twists. A Pretzel link is a knot if and only if [34]:

1. One \(q_i\) is even, \(q_j\) is odd for \(j\neq i\).

2. All \(q_i\) are odd, \(n\) is odd.

As usual, the sign of \(q_i\) determines if the unoriented twist is either positive or negative. Choosing an orientation for \(K\), the following proposition lists the possible decompositions of \(K\) as a tangle in terms of codirected braids or alternating braidings.

Proposition 33 shows that the Alexander module of a pretzel knot can be expressed in terms of the matrix product \(PDQ\), where \(P\) and \(Q\) are the \(n \times 2n\) and \(2n \times n\) matrices, respectively, given by \[P = \begin{pmatrix} -1 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\ 0 & 1 & -1 & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -1 & \cdots & 0 & 0 & 0 \\ \vdots & & & & & & & & \vdots \\ 0 & 0 & 0 & 0 & 0 & \cdots & 1 & -1 & 0 \end{pmatrix}, \qquad Q = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ 0 & 0 & 0 & \cdots & 1 \\ 1 & 0 & 0 & \cdots & 0 \end{pmatrix}.\] Here \(D\) denotes a block diagonal matrix whose diagonal blocks are \(2\times 2\) matrices determined by the basic spans that corresponding to the alternating and codirected braidings of the pretzel tangle \(T\). Note that the elementary row and column operations defined by the pretzel closure matrices \(P\) and \(Q\) only involve adjacent rows and columns if we regard indices modulo \(n\), so that the first and the last column are adjacent too, matching the cyclic nature of a pretzel knot. Consequently, we are able to explicitly describe the linear map \({\mathcal{A}}_t(K)=t_1\circ D \circ t_2\), which has a cyclic tridiagonal structure: its associated matrix is tridiagonal with non-zero elements at its corners.

Let \(S\) be the cycle \(\sigma=(1,\ldots,n)\), and let \(M_{[i]}\) be the principal minor of \(M\) that is obtained when removing the \(i\)-th row and column of \(M\). Since the resulting matrix is tridiagonal, \(M_{[i]}\) is the continuant \(K^{i}_{n-1}\) defined recursively as in 19 : \[\label{eqn:opencontinuantgeneral} K^{i}_0=1, \quad K_1^{i}=a_{\sigma^{i}(1)}, \quad K_m^{i}=a_{\sigma^i(m)}K^{i}_{m-1}-u_{\sigma^i(m-1)}l_{\sigma^i(m)}K^{i}_{m-2}, \quad m \geq 2.\tag{21}\] Since \(M\) presents the Alexander module of the pretzel knot \(K\), we know that any minor of size \(n-1\) leads to the Alexander polynomial, since the first elementary ideal is principal. Then \[M_{[i+1]}(q_1,\ldots,q_n;t)=M_{[i]}(\sigma(q_1,\ldots,q_n);t),\] which also matches the fact that pretzel knots are isotopic under cyclic permutations of the indices, that is, \[K(q_1,\ldots,q_n)\sim K(\sigma(q_1,\ldots,q_n)).\] We will choose the minor \(M_{[1]}\) so that the recursive definition and the first diagonal entry of the resulting continuant \(K\) starts involving \(q_1\) and \(q_2\). Hence

We apply Proposition 35 to each of the possible pretzel knots listed in Proposition 32.

5.1 Odd pretzel knots \(K=P(2k_1+1,\ldots, 2k_{n}+1)\)↩︎

In this case, \(n\) is odd and every braiding of \(K\) is an alternating braiding, and moreover, we may choose orientations so that the first elementary tangle of each braiding is \(R^3(X_{\pm})\). Then \[\tilde{f}_-^{2k_i+1}(t^{-1})=f_+(t)+k_i(1-t)h, \quad D_i=\begin{pmatrix} (1+k_i)(1-t) & t -k_i(1-t) \\ 1+k_i(1-t) & -k_i(1-t)\end{pmatrix}.\] Applying Proposition 35, we get that \[a_i=(-1-k_i-k_{i+1})(1-t), \quad l_i=-1-k_i(1-t), \quad u_i=t-k_{i+1}(1-t).\] Note that \(a_i=l_i+u_i\) for all \(i\), hence \(\Delta_K(t)\) equals \[{!}{ \det T_{n-1}= \begin{vmatrix} (-1-k_1-k_2)(1-t) & t-k_2(1-t) & 0 & \ldots & 0 & 0 \\ -1-k_2(1-t) & (-1-k_2-k_3)(1-t) & t-k_3(1-t) & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & -1-k_{n-1}(1-t) & (-1-k_{n-1}-k_n)(1-t) \end{vmatrix} }\] The structure of the tridiagonal matrix \(T_n\) leads us to prove the following result regarding the zeros of its Alexander polynomial.

We can also use Proposition 35 to obtain a closed formula for the Alexander polynomial of an odd pretzel knot, recently proved by Belousov [16] using the skein relation:

5.2 Even pretzel knots \(K(2k_1,2k_2+1,\ldots,2k_n+1)\)↩︎

Throughout this section, let us write \(q_1=2k_1\), \(q_i:=2k_i+1\) for \(i\geq 2\). In this case, there are two possible braidings according to Proposition 32, so we subdivide our computations according to the parity of \(n\).

5.2.1 \(n=2p\)↩︎

All braidings that form the pretzel knot are codirected and their direction alternates: we may choose an orientation for \(K\) such that the first elementary tangle of the braid corresponding to \(2k_1\) is \(X_{\pm}\). As a consequence, for \(i=1,\ldots,p,\), Corollary 15 implies that \[\label{eqn:Diforevenpr} D_{2i-1}=\begin{pmatrix} 1-[q_{2i-1}]_{-t^{-1}} & [q_{2i-1}]_{-t^{-1}} \\ t^{-1}[q_{2i-1}]_{-t^{-1}} & 1-t^{-1}[q_{2i-1}]_{-t^{-1}} \end{pmatrix}, \quad D_{2i}=\begin{pmatrix} 1-[q_{2i}]_{-t} & [q_{2i}]_{-t} \\ t[q_{2i}]_{-t} & 1-t[q_{2i}]_{-t} \end{pmatrix},\tag{25}\] thus direct application of Proposition 35 yields that \[a_{2i-1}=-t^{-1}[q_{2i-1}]_{-t^{-1}}+[q_{2i}]_{-t}, \quad u_{2i-1}=[q_{2i}]_{-t}, \quad l_{2i-1}=-t^{-1}[q_{2i-1}]_{-t^{-1}},\] and also \[a_{2i}=-t[q_{2i}]_{-t}+[q_{2i+1}]_{-t^{-1}}, \quad u_{2i}=[q_{2i+1}]_{-t^{-1}}, \quad l_{2i}=-t[q_{2i}]_{-t}.\] Note that for all \(i=1,\ldots, n\), \(a_i=l_i+u_i\). The next proposition provides a closed formula for the Alexander polynomial in terms of quantum integers.

Equation ?? allows us to prove that the zeros of the Alexander polynomial are bounded when \(k_i>0\). We prove first an elementary lemma:

As the determinant of the knot is a non-zero odd integer, \(t=-1\) is not a root of the Alexander polynomial:

5.2.2 \(n=2p+1\)↩︎

In this case, the first braiding is alternating, and the others are codirected. Proceeding as before, we may orient \(K\) so that the first braiding is exactly \(\widetilde{T}_+^{2k_1}\), \[D_1=\mathop{\mathrm{Id}}-k_1(1-t^{-1})h=\begin{pmatrix} 1-k_1(1-t^{-1}) & k_1(1-t^{-1}) \\-k_1(1-t^{-1}) & 1+k_1(1-t^{-1})\end{pmatrix},\] where we recall that \(q_1=2k_1\). \(D_i\), for \(i\geq 2\), matches exactly the pattern from the case \(n=2p\) in 25 . As a consequence, the same recursion is satisfied, with the sole exception of the initial condition that equals \[a_1=k_1[2]_{-t^{-1}}+[q_2]_{-t}, \quad u_1=[q_2]_{-t}, \quad l_1=k_1[2]_{-t^{-1}}.\] The same strategy yields an analogous formula for even pretzel knots with and odd number of strands, as the next proposition shows.

In this case, it is not longer true that any root \(t\) of the Alexander polynomial of \(K\) satisfies \(\abs{t}\leq 1\). For example, the Alexander polynomial of the alternating knot \(9_{11}=P(-5,2,1,1,1)\) equals \(\Delta_K(t)=1-5t+7t^2-7t^3+7t^4-5t^5+t^6\), which has a positive real root in \([3,4]\). We provide an alternative proof to [22] of Hoste’s bound for odd \(n\) for this family:

6 The functor \({\mathcal{A}}_{-1}\)↩︎

The functor \({\mathcal{A}}_{-1}\) is related to several classical invariants of knots and encodes several aspects of their arithmetic and topology. For instance, it is a classical result that evaluating the Alexander polynomial at \(t=-1\) yields the determinant of \(K\), defined as the order of \(H_1(X_2)\), where \(X_2\) denotes the double cover of \(S^3\) branched over \(K\). Another well-known case where the specialization \(t=-1\) recovers an important invariant arises in the theory of Alexander linear quandles: when \(t=-1\), the quandle operation on \({\mathbb{Z}}_p\) reduces to \[a \triangleright_{-1} b = 2a - b,\] which is classically connected to knot colorings [35].

In our setting, \(t=-1\) is a fixed point of the involution \(\tau:{\mathbb{C}}^{\ast}\to{\mathbb{C}}^{\ast}\), \(\tau(t)=t^{-1}\). Geometrically, Remark 13 shows that the fibres of \({\mathcal{A}}(T)\) and \({\mathcal{A}}(T')\) at \(t=-1\) are isomorphic, a fact which also follows algebraically from Corollary 11. Consequently, \({\mathcal{A}}_{-1}(T)\) depends only on the unoriented isotopy class of the tangle, which we denote by \([T]\). From a categorical perspective, the functor \({\mathcal{T}}_{-1}\) factors through the forgetful functor \({\mathcal{T}}\to {\mathcal{T}}^{\mathrm{unor}}\), where \({\mathcal{T}}^{\mathrm{unor}}\) denotes the category of unoriented tangles. Thus:

Moreover, our computations show that \(t=-1\) is the unique value for which the maps \(f_{\pm} : {\mathbb{C}}^2 \to {\mathbb{C}}^2\) fail to be diagonalizable, as the spectrum of \(f_{\pm}\) is given by \(\sigma=\lbrace1,-t \rbrace\). In addition, since \([n]_{-1}=n\), Propositions 14 and 18 specialize to \[\begin{align} f^n_{+}(a_1,a_2) & =((1+n)a_1-na_2, na_1+(1-n)a_2)=\tilde{f}_-^n(a_1,a_2) \\f^n_{-}(a_1,a_2) & =((1-n)a_1+na_2, -na_1+(1+n)a_2)=\tilde{f}_+^n(a_1,a_2), \end{align}\] which leads us to the following definition.

It is immediate to verify that \(f^{s'} \circ f^s= f^{s'+s}\) and that \(f=f_+, \,f^{-1}=f_-\) . As we did before, we will in this section identify a linear map with a basic span via the embedding of \(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}}\) inside \(\mathop{\mathrm{Span}}(\mathop{\mathrm{\mathbf{Vect}}}_{{\mathbb{C}}})\). A direct computation of the rotation of \(f^s\) shows that they behave nicely with respect to rotations:

As a consequence, following the conventions from [27], where \([X_+]=[-1]\) and \([X_-]=[+1]\), we get that \({\mathcal{A}}_{-1}([n])=f^{-1/n}\) for \(n\in {\mathbb{Z}}\). Thus

For any rational number \(p/q\in \mathbb{Q}\), there exists a positive standard continued fraction \([a_1,\ldots,a_n]\), which is unique if all \(a_i\) have the same sign and \(n\) is odd. In analogy, rational tangles admit a continued fraction description: every rational tangle is isotopic to a tangle obtained by iterated vertical compositions of the elementary tangles \([X_{\pm}]\) on the top, together with successive horizontal additions of \([X_{\pm}]\) on the right. Moreover, there exists a canonical continued fraction form of the tangle, characterized by the property that its planar projection is alternating and the number of iterations is odd, topologically mirroring the arithmetic properties of continued fractions.

We restate this theorem from [27] in terms of rotations and compositions:

6.1 Colorings↩︎

The fraction \(p/q\) associated with a rational tangle can be obtained via colorings. We briefly summarize the relevant results and definitions here, referring to [27] for a detailed exposition and complete proofs.

Given any knot \(K\) represented in terms of a planar diagram, an (integral) coloring of \(K\) is a function from the set of arcs of \(K\) to \(\mathbb{Z}\), such that Fox’s coloring equations \[\label{eqn:Foxcoloreq} 2a=b+c,\tag{29}\] are satisfied at any crossing, where \(a\) denotes the color of the overcrossing arc and \(b,c\) are the colors of the two undercrossing arcs.

Unlike knots, where (non-trivial) colorings must sometimes be reduced to \(\mathbb{Z}_n\) in order to exist, rational tangles are integrally colorable: one can choose colors for the initial strands of any rational knot (where the tangle starts to wind), so that the coloring propagates to the rest of the tangle.

To any coloring of a rational tangle \(T\), as a 2-tangle, we can associate its color matrix, \[M(T)=\begin{pmatrix} a & b \\c & d \end{pmatrix},\] where \(a,b,c,d \in \mathbb{Z}\) are the colors of the NW, NE, SW and SE external strands of \(T\), respectively. For example, the color matrix of the coloring displayed in Figure 8 is \(M=\left( \begin{smallmatrix} 16 & 23 \\ 0 & 7 \end{smallmatrix} \right)\).

We also recall the following properties of any color matrix of \(T\):

• \(a-b=c-d\) (defined in [27] as the diagonal sum rule)

• For any \(k,l\in {\mathbb{Z}}\), \(\left( \begin{smallmatrix} ka+l & kb+l \\ kc+l & kd+l\end{smallmatrix} \right)\) is also a coloring matrix for a rational tangle \(T\).

Thus any coloring of \(T\) defines a coloring matrix \(M(T)\). Conversely, by the propagation property, any coloring matrix \(M(T)\) uniquely recovers the full coloring on the tangle \(T\). For this reason, we shall occasionally refer to \(M(T)\) itself as a coloring of \(T\). The coloring invariant that can be associated to any rational tangle \(T\) is the rational number \[f(T)=\frac{b-a}{b-d} \in \mathbb{Q},\] which classifies the rational tangle \(T\) up to isotopy. It coincides with the “topological” fraction that can be associated to any rational tangle from its continued fraction form.

Colorings are classically related to Alexander quandles [32], [35], since Fox coloring equation 29 is the specialization at \(t=-1\) of the Alexander quandle relation 4 at every crossing. The goal of this section is to provide a complex-geometrical interpretation in terms of spans of the fraction \(p/q\) of a rational tangle, relating colorings and the functor \({\mathcal{A}}_{-1}\). We first restate and prove some of the previous properties and statements in terms of representations, quandles and the functor \({\mathcal{A}}_{-1}\).

In what follows, \(T\) is assumed to be a rational tangle, and \(\partial T=\epsilon \sqcup \varepsilon\).

First of all, \({\mathcal{R}}_{{\mathrm{AQ}_{-1}}}(\epsilon)\cong R_{{\mathrm{AQ}_{-1}}}(\varepsilon)\cong {\mathbb{C}}^2\), as \(Q_{\epsilon}\cong Q_{\varepsilon}\) is the free quandle in two generators. \({\mathcal{R}}_{{\mathrm{AQ}_{-1}}}(T)\) is also two-dimensional for any rational tangle \(T\):

The diagonal sum rule property of colorings of rational tangles can also be interpreted in terms of representations, as the next proposition shows.

The following proposition clarifies its relation with the functor \({\mathcal{A}}_{-1}(T)\).

As a consequence, the span \({\mathcal{A}}_{-1}(T)\) can be regarded as a monomorphism \(\phi_T:=(f,g):{\mathbb{C}}^2 \longrightarrow {\mathbb{C}}^4\) that satisfies that \(\phi_T(1,1)=(1,1,1,1)\). The image of \(\phi_T\) is a plane in \({\mathbb{C}}^4\) generated by \(v_0:=(1,1,1,1)\) and any non-trivial coloring \(v:=(a,b,c,d)\) on \(\partial{T}\). Note that the choice of any equivalent span to \({\mathcal{A}}_{-1}(T)\) corresponds to a different choice of basis for \({\mathcal{R}}_{{\mathrm{AQ}_{-1}}}(T)\cong {\mathbb{C}}^2\), but the plane defined by the image of \(\phi_T\) remains invariant. That is,

Let us recall that the complex grassmannian \(\mathop{\mathrm{Gr}}(2,4)\) can be realized as a complex algebraic subvariety of \(\mathbb{P}^5({\mathbb{C}})\) via the Plücker embedding, that maps any \(P=\langle u,v\rangle \in \mathop{\mathrm{Gr}}(2,4)\) to the line generated by the bivector \(u\wedge v\) in \(\bigwedge^2 \mathbb{C}^4\), a point in \(\mathbb{P}(\bigwedge^2 \mathbb{C}^4) \cong \mathbb{P}^5(\mathbb{C})\). Its image in \(\mathbb{P}^5(\mathbb{C})\) is the Plücker quadric \(\mathcal{C}\) defined by the equation \[p_{12}p_{34}-p_{13}p_{24}+p_{14}p_{23}=0,\] where \(p_{ij}=\begin{vmatrix} v_i & v_j \\w_i & w_j \end{vmatrix}\) are the standard coordinates of \(\bigwedge^2 {\mathbb{C}}^4\).

We are interested in describing the set \[{\mathcal{A}}_{-1}^{rat}:=\lbrace {\mathcal{A}}_{-1}(T) : T \text{ is a rational tangle} \rbrace,\] which by Corollary 57 can be understood as the image (via colorings) of all rational tangles inside \(G(2,4)\), thanks to the functor \({\mathcal{A}}_{-1}\). First of all, let us consider the Schubert variety \(\sigma \subset \mathcal{C}\) defined as \[\sigma = \lbrace P\in \mathcal{C} \mid \dim( \langle v_0\rangle \cap P ) \geq 1\rbrace= \lbrace \langle v_0,v\rangle \mid v\in {\mathbb{C}}^4 \rbrace \subset \mathcal{C} \subset \mathop{\mathrm{Gr}}(2,4),\] which consists of planes in \({\mathbb{C}}^4\) that contain the line generated by \(v_0\). We note that in this case \(p_{ij}= \begin{vmatrix} 1 & 1 \\ v_i & v_j \end{vmatrix}=v_j-v_i\), so that \(\sigma\) is defined by the three linear equations on \(\mathop{\mathrm{Gr}}(2,4)\) \[\begin{align} p_{12}-p_{13}+p_{23} & = 0, \tag{30} \\ p_{12}-p_{14}+p_{24} & = 0, \tag{31}\\ p_{13}-p_{14}+p_{34} & =0.\tag{32} \end{align}\] Thus \(\sigma \cong \mathbb{P}^2({\mathbb{C}})\), parametrized by \([p_{12}:p_{13}:p_{14}]\).

Furthermore, Proposition 55 establishes that any point \(P_T\) in \(G(2,4)\), which arises from a rational tangle \(T\), satisfies the additional equation \(v_1-v_2=v_3-v_4\). This imposes the additional linear equation \(p_{12}=p_{34}\) on \(\sigma\), equivalent to \[\label{eqn:plucker4} p_{12}+p_{13}-p_{14}=0.\tag{33}\] We summarize this discussion in the following theorem.

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