A diagrammatic approach to the AJ Conjecture

TL;DR

This paper introduces a diagram-dependent version of the $ ilde{A}$-polynomial for knots, proves it satisfies one direction of the AJ Conjecture using octahedral decompositions and state sum formulas, advancing understanding of knot invariants.

Contribution

It proposes a new diagram-dependent $ ilde{A}$-polynomial and proves it satisfies one side of the AJ Conjecture using geometric and algebraic methods.

Findings

The diagram-dependent $ ilde{A}$-polynomial aligns with the $ ilde{A}$-polynomial conjecturally.

The proof employs octahedral decompositions from planar knot diagrams.

The approach connects geometric decompositions with quantum invariants.

Abstract

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $\hat{A}$ polynomial), with a classical invariant, namely the defining polynomial $A$ of the $\psl$ character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the $\hat{A}$-polynomial (after we set $q=1$, and excluding those of $L$-degree zero) coincides with those of the $A$-polynomial. In this paper, we introduce a version of the $\hat{A}$-polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the $\hat{A}$-polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the $R$-matrix state sum formula for the colored Jones polynomial, and its certificate.

Equations

$$\sum _{j=0}^d{c}_j(q,q^n)J_K(n+j)=0,\text{for all}n\in \text{mathbbm}N,$$

$$(Ef)(n)=f(n+1),(Qf)(n)=q^{n}f(n).$$

$$\mathrm{Ann}(f)=\{P\in \text{mathbbm}Z[q,Q]⟨E⟩|Pf=0\}$$

$$J_K(n)=\sum _{\text{mathbbm}{Z}^{c(D)+1}}{w}_D(n,k)$$

$$f(n)=\sum _{k\in \text{mathbbm}{Z}^r}F(n,k)$$

$$\mathrm{Ann}(F)\subset \text{mathbbm}Q[q,Q,Q_k]⟨E,E_k⟩$$

$$\phi :\text{mathbbm}Q[q,Q]⟨E,E_k⟩\to \text{mathbbm}Q[q,Q]⟨E⟩,\phi (E_{k_i})=1,i=1,\dots ,r.$$

$$\phi (\mathrm{Ann}(F)\cap \text{mathbbm}Q[q,Q]⟨E,E_k⟩)\subset \mathrm{Ann}(f)$$

$$(z,z^{\prime },z^{\prime \prime })=\left(z,\frac{1}{1-z},1-\frac{1}{z}\right)$$

$$z{z}^{\prime }{z}^{\prime \prime }=-1,z{z}^{\prime }-z+1=0$$

$$\prod _{\Delta \in N(e)}{z}_{\Delta }=1,$$

$${\mathcal{G}}_{\mathcal{T}}\to \overline{X}(M,\mathrm{PSL}(2,\text{mathbbm}C)).$$

$$(h_{\mu },h_{\lambda }):\overline{X}(M,\mathrm{PSL}(2,\text{mathbbm}C))\to \text{mathbbm}{C}^{\ast }\times \text{mathbbm}{C}^{\ast }.$$

$$h_{\gamma }=\prod _{leftsegments}{z}_{left}\prod _{rightsegments}{z}_{right}^{-1}.$$

$$\prod _{c\in \text{corners}(R)}{z}_c=1.$$

$$h_{{\gamma }_i}=-\prod _{c\text{left corner}}{z}_c=-\prod _{c\text{right corner}}{z}_c^{-1},$$

$$h_{\gamma }=\prod _{i\in I}{h}_{{\gamma }_i}.$$

$$w{z}_{ui}{z}_{uo}=1,w{z}_{li}{z}_{lo}=1.$$

$$z_{1,lo}{z}_{2,ui}^{\prime }{z}_{2,uo}^{\prime \prime }\dots z_{n-1,ui}^{\prime }{z}_{n-1,uo}^{\prime \prime }{z}_{n,li}{z}_{n-1,uo}^{\prime }{z}_{n-1,ui}^{\prime \prime }\dots z_{2,uo}^{\prime }{z}_{2,ui}^{\prime \prime }=1.$$

$$z_{n,lo}=z_{1,lo}{w}_n^{-1}\underset{j=2}{\prod ^{n-1}}{w}_j,z_{n,li}=z_{1,li}{w}_1\underset{j=2}{\prod ^{n-1}}{w}_j^{-1}.$$

$$z_{n,ui}=z_{1,ui}{w}_1\underset{j=2}{\prod ^{n-1}}{w}_j^{-1},z_{n,uo}=z_{1,uo}{w}_n^{-1}\underset{j=2}{\prod ^{n-1}}{w}_j.$$

$$z_{n,li}{z}_{1,lo}=\underset{j=2}{\prod ^{n-1}}{z}_{j,ui}{z}_{j,uo}$$

$$z_{n,li}{z}_{1,lo}=\underset{j=2}{\prod ^{n-1}}{w}_j^{-1}$$

$$z_{1,uo}{z}_{2,li}^{\prime \prime }{z}_{2,lo}^{\prime }\dots z_{n-1,li}^{\prime \prime }{z}_{n-1,lo}^{\prime }{z}_{n,ui}{z}_{n-1,lo}^{\prime \prime }{z}_{n-1,li}^{\prime }\dots z_{2,lo}^{\prime \prime }{z}_{2,li}^{\prime }=1,$$

$$\prod _{v\text{corner of}{R}_i}f(v)=1$$

$${\mathcal{G}}_D=\{(w_c,z_{c,ui},z_{c,uo},z_{c,li},z_{c,lo}{)}_{c\in c(D)}\in ({\mathbb{C}}^{\ast \ast }{)}^{5c(D)}|t_i=1,s_j=1,r_k=1\}.$$

$$h_{\mu }=-z_{1,lo}{z}_{2,ui}^{\prime }{z}_{2,ui}^{\prime \prime }.$$

$$w_{\mu }=h_{\mu }=\frac{z_{1,lo}}{z_{2,ui}}.$$

$$h_a=-\prod _{\text{c left corner}}{z}_c$$

$$h_a=-\prod _{\text{c right corner}}{z}_c^{-1}.$$

Full text

1991 Mathematics Classification. Primary 57N10. Secondary 57M25.

*Key words and phrases: Knot, planar projection, planar diagram, Jones polynomial, colored Jones polynomial, AJ Conjecture, $q$-holonomic sequences, certificate, holonomic modules, gluing equations, character variety. *

1. 1 Introduction

2. 1.1 The colored Jones polynomial and the AJ Conjecture

3. 1.2 $q$-holonomic sums

4. 1.3 Our results

5. 1.4 Sketch of the proof

6. 2 Knot diagrams, their octahedral decomposition and their gluing equations

7. 2.1 Ideal triangulations and their gluing equations

8. 2.2 Spines and gluing equations

9. 2.3 The octahedral decomposition of a knot diagram

10. 2.4 The spine of the $5T$-triangulation of a knot diagram and its gluing equations

11. 2.5 Labeled knot diagrams

12. 2.6 Analysis of triangle and shingle relations

13. 2.7 Analysis of big region equations

14. 2.8 Formulas for the loop equations

15. 2.9 A square root of the holonomy of the longitude

16. 3 $q$-holonomic functions, creative telescoping and certificates

17. 4 The colored Jones polynomial of a knot

18. 4.1 State sum formula for the colored Jones polynomial of a knot diagram

19. 4.2 The annihilator ideal of the summand of the state sum

20. 5 Matching the annihilator ideal and the gluing equations

21. 5.1 From the annihilator of the state summand to the gluing equations variety

22. 5.2 Proof of Theorem 1.1

1. Introduction

1.1. The colored Jones polynomial and the AJ Conjecture

The Jones polynomial of a knot [Jon87] is a powerful knot invariant with deep connections with quantum field theory, discovered by Witten [Wit89]. The discoveries of Jones and Witten gave rise to Quantum Topology. An even more powerful invariant is the colored Jones polynomial $J_{K}(n)\in\mathbbm{Z}[q^{\pm 1}]$ of a knot $K$, a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. Since the dependence of the colored Jones polynomial $J_{K}(n)$ on the variable $q$ plays no role in our paper, we omit it from the notation. The colored Jones polynomial determines the Alexander polynomial [BNG96], is conjectured to determine the volume of a hyperbolic knot [Kas95, Kas97, MM01], is conjectured to select two out of finitely many slopes of incompressible surfaces of the knot complement [Gar11b], and is expected to determine the $\mathrm{SL}(2,\mathbbm{C})$ character variety of the knot, viewed from the boundary [Gar04]. The latter is the AJ Conjecture, which is the focus of our paper.

The starting point of the AJ Conjecture [Gar04] is the fact that the colored Jones polynomial $J_{K}(n)$ of a knot $K$ is $q$-holonomic [GL05], that is, it satisfies a nontrivial linear recursion relation

[TABLE]

where $c_{j}(u,v)\in\mathbbm{Z}[u,v]$ for all $j$. We can write the above equation in operator form as follows $PJ_{K}=0$ where $P=\sum_{j}c_{j}(q,Q)E^{j}$ is an element of the ring $\mathbbm{Z}[q,Q]\langle E\rangle$ where $EQ=qQE$ are the operators that act on sequences of functions $f(n)$ by:

[TABLE]

Observe that the set

[TABLE]

is a left ideal of $\mathbbm{Z}[q,Q]\langle E\rangle$, nonzero when $f$ is $q$-holonomic. Although the latter ring is not a principal left ideal domain, its localization $\mathbbm{Q}(q,Q)\langle E\rangle$ is, and cleaning denominators allows one to define a minimal $E$-order, content-free element $\hat{A}_{K}(q,Q,E)\in\mathbbm{Z}[q,Q]\langle E\rangle$ which annihilates the colored Jones polynomial.

On the other hand, the $A$-polynomial of a knot [CCG*+*94] $A_{K}(L,M)\in\mathbbm{Z}[L,M]$ is the defining polynomial for the character variety of $\mathrm{SL}(2,\mathbbm{C})$ representations of the boundary of the knot complement that extend to representations of the knot complement.

The AJ Conjecture asserts that the irreducible factors of $\hat{A}_{K}(1,Q,E)$ of positive $E$-degree coincide with those of $A_{K}(Q,E^{-2})$. The AJ Conjecture is known for most 2-bridge knots, and some 3-strand pretzel knots; see [L0̂6] and [LZ17].

Let us briefly now discuss the $q$-holonomicity of the colored Jones polynomial [GL05]: this follows naturally from the fact that the latter can be expressed as a state-sum formula using a labeled, oriented diagram $D$ of the knot, placing an $R$-matrix at each crossing and contracting indices as described for instance in Turaev’s book [Tur94]. For a diagram $D$ with $c(D)$ crossings, this leads to a formula of the form

[TABLE]

where the summand $w_{D}(n,k)$ is a $q$-proper hypergeometric function and for fixed $n$, the support of the summand is a finite set. The fundamental theoreom of $q$-holonomic functions of Wilf-Zeilberger [WZ92] concludes that $J_{K}(n)$ is $q$-holonomic. Usually this ends the benefits of (4), aside from its sometimes use as a means of computing some values of the colored Jones polynomial for knots with small (eg $12$ or less) number of crossings and small color (eg, $n<10$).

Aside from quantum topology, and key to the results of our paper, is the fact that a planar projection $D$ of a knot $K$ gives rise to an ideal octahedral decomposition of its complement minus two spheres, and thus to a gluing equations variety $\mathcal{G}_{D}$ and to an $A$-polynomial $A_{D}$ reviewed in Section 2 below. In [KKY18], Kim-Kim-Yoon prove that $A_{D}$ coincides with the $A$-polynomial of $K$, and in [KP] Kim-Park prove that $\mathcal{G}_{D}$ is, up to birational equivalence, invariant under Reidemeister moves, and forms a diagrammatic model for the decorated $\mathrm{PSL}(2,\mathbbm{C})$ character variety of the knot.

The aim of the paper is to highlight the fact that formulas of the form (4) lead to further knot invariants which are natural from the point of view of holonomic modules and form a rephrasing of the AJ Conjecture that connects well with the results of [KKY18] and [KP].

1.2. $q$-holonomic sums

To motivate our results, consider a sum of the form

[TABLE]

where $n\in\mathbbm{Z}$ and $k=(k_{1},\dots,k_{r})\in\mathbbm{Z}^{r}$ and $F(n,k)$ is a proper $q$-hypergeometric function with compact support for fixed $n$. Then $f$ is $q$-holonomic but more is true. The annihilator

[TABLE]

of the summand is a $q$-holonomic left ideal where $E_{k}=(E_{k_{1}},\dots,E_{k_{r}})$ and $Q_{k}=(Q_{k_{1}},\dots,Q_{k_{r}})$ are operators, each acting in one of the $r+1$ variables $(n,k)$ with the obvious commutation relations (operators acting on different variables commute and the ones acting on the same variable $q$-commute). Consider the map

[TABLE]

It is a fact (see Proposition 3.2 below) that

[TABLE]

and that the left hand side is nonzero. Elements of the left hand side are usually called “good certificates”, and in practice one uses the above inclusion to compute a recursion for the sum [PWZ96, Zei91]. If $\hat{A}^{c}_{F}(q,Q,E)$ and $\hat{A}_{f}(q,Q,E)$ denotes generators of the left and the right hand side of (7), it follows that $\hat{A}_{f}(q,Q,E)$ is a right divisor of $\hat{A}^{c}_{F}(q,Q,E)$. We will call the latter the certificate recursion of $f$ obtained from (5).

In a sense, the certificate recursion of $f$ is more natural than the minimal order recursion and that is the case for holonomic $D$-modules and their push-forward, discussed for instance by Lairez [Lai16].

What is more important for us is that if one allows presentations of $f$ of the form (5) where $F$ is allowed to change by for instance, consequences of the $q$-binomial identity, then one can obtain an operator $\hat{A}^{c}_{f}(q,Q,E)$ which is independent of the chosen presentation.

1.3. Our results

Applying the above discussion to (4) with $F=w_{D}$, allows us to introduce the certificate recursion $\hat{A}_{D}^{c}(q,Q,E)\in\mathbbm{Z}[q,Q]\langle E\rangle$ of the colored Jones polynomial, which depends on a labeled, oriented planar diagram $D$ of a knot. We can also define $\hat{A}_{K}^{c}(q,Q,E)\in\mathbbm{Z}[q,Q]\langle E\rangle$ to be the left gcd of the elements $\hat{A}_{D}^{c}$ in the local ring $\mathbbm{Q}(q,Q)\langle E\rangle$, lifted back to $\mathbbm{Z}[q,Q]\langle E\rangle$.

We now have all the ingredients to formulate one direction of a refined AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the $R$-matrix state sum formula for the colored Jones polynomial, and its certificate.

Theorem 1.1**.**

*For every knot $K$,

(a) $\hat{A}_{K}$ divides $\hat{A}_{K}^{c}$.

(b) Every irreducible factor of $A_{K}(Q,E^{-2})$ of positive $E$-degree is a factor of $\hat{A}_{K}^{c}(1,Q,E)$.*

Remark 1.2**.**

The $\hat{A}_{K}$-polynomial has only been computed in a handful of cases, see [GS10], [GM11], [GK12] and [GK13]. In all cases where $\hat{A}_{K}$ is known, it is actually obtained from certificates and in that case $\hat{A}_{K}^{c}=\hat{A}_{K}$.

Question 1.3**.**

Is it true that for any knot $K$, one has $\hat{A}_{K}^{c}=\hat{A}_{K}$ ?

Question 1.4**.**

Is it true that the certificate recursion $\hat{A}^{c}_{D}$ of a planar projection of a knot is invariant under Reidemester moves on $D$?

A positive answer to the latter question is a quantum analogue of the fact that the gluing equation variety $\mathcal{G}_{D}$ associated to a diagram $D$ is independent of $D$, a result that was announced by Kim and Park [KP]. We believe that the above question has a positive answer, coming from the fact that the Yang-Baxter equation for the R-matrix follows from a $q$-binomial identity, but we will postpone this investigation to a future publication.

1.4. Sketch of the proof

To prove Theorem 1.1, we fix a planar projection $D$ of an oriented knot $K$. On the one hand, the planar projection gives rise to an ideal decomposition of the knot complement (minus two points) using one ideal octahedron per crossing, subdividing further each octahedron to five ideal tetrahedra. This ideal decomposition gives rise to a gluing equations variety, discussed in Section 2. On the other hand, the planar projection gives a state-sum for the colored Jones polynomial, by placing one $R$-matrix per crossing and contracting indices. The summand of this state-sum is $q$-proper hypergeometric and its annihilator defines an ideal in a quantum Weyl algebra, discussed in Section 4. The annihilator ideal is matched when $q=1$ with the gluing equations ideal in the key Proposition 5.1. This matching, implicit in the Grenoble notes of D. Thurston [Thu99], combined with a certificate (which is a quantum version of the projection map from gluing equations variety to $\mathbbm{C}^{*}\times\mathbbm{C}^{*}$), and with the fact that the gluing equation variety sees all components of the $\mathrm{PSL}(2,\mathbbm{C})$ character variety [KKY18], conclude the proof of our main theorem.

Our method of proof for Theorem 1.1 using certificates to show one direction of the AJ Conjecture is general and flexible and can be applied in numerous other situations, in particular to a proof of one direction of the AJ Conjecture for state-integrals, and to one direction of the AJ Conjecture for the 3Dindex [AM, Dim13]. This will be studied in detail in a later publication. For a discussion of the AJ Conjecture for state-integrals and for a proof in the case of the simplest hyperbolic knot, see [AM].

Finally, our proof of Theorem 1.1 does not imply any relation between the Newton polygon of the $\hat{A}_{K}(q,Q,E)$ polynomial and that of $A_{K}(1,Q,E)$. If the two Newton polygons coincided, the Slope Conjecture of [Gar11b] would follow, as was explained in [Gar11a]. Nonetheless, the Slope Conjecture is an open problem.

2. Knot diagrams, their octahedral decomposition and

their gluing equations

2.1. Ideal triangulations and their gluing equations

Given an ideal triangulation $\mathcal{T}$ of a 3-manifold $M$ with cusps, Thurston’s gluing equations (one for each edge of $\mathcal{T}$) give a way to describe the hyperbolic structure on $M$ and its deformation if $M$ is hyperbolic [Thu77, NZ85]. The gluing equations define an affine variety $\mathcal{G}_{\mathcal{T}}$, the so-called gluing equations variety, whose definition we now recall. The edges of each combinatorial ideal tetrahedron get assigned variables, with opposite edges having the same variable as in the left hand side of Figure 1. The triple of variables (often called a triple of shapes of the tetrahedron)

[TABLE]

satisfies the equations

[TABLE]

and every solution of (8) uniquely defines a triple of shapes of a tetrahedron. Note that the shapes of the tetrahedron $z$, $z^{\prime}$, or $z^{\prime\prime}$ lie in $\mathbbm{C}^{**}=\mathbbm{C}{\smallsetminus}\{0,1\}$, and that they are uniquely determined by $z\in\mathbbm{C}^{**}$. When we talk about assigning a shape $z$ to a tetrahedron below, it determines shapes $z^{\prime}$ and $z^{\prime\prime}$ as in Figure 1.

Given an ideal triangulation $\mathcal{T}$ with $N$ tetrahedra, assign shapes $z_{i}$ for $i=1,\dots,N$ to each tetrahedron. If $e$ is an edge of $\mathcal{T}$ the corresponding gluing equation is given by

[TABLE]

where $N(e)$ is the set of all tetrahedra that meet along the edge $e$, and $z_{\Delta}$ is the shape parameter corresponding to the edge $e$ of $\Delta$. The gluing equation variety $\mathcal{G}_{\mathcal{T}}$ is the affine variety in the variables $(z_{1},\dots,z_{N})\in(\mathbbm{C}^{**})^{N}$ defined by the edge gluing equations, for all edges of $\mathcal{T}$. Equivalently, it is the affine variety in the variables $(z_{1},z_{1}^{\prime},z_{1}^{\prime\prime},\dots,z_{N},z_{N}^{\prime},z_{N}^{\prime\prime})\in\mathbbm{C}^{3N}$ defined by the edge equations and the equations (8), one for each tetrahedron.

We next discuss the relation between a solution to the gluing equations and decorated (or sometimes called, augmented) $\mathrm{PSL}(2,\mathbbm{C})$ representations of the fundamental group of the underlying 3-manifold $M$. The construction of decorated representations from solutions to the gluing equations appears for instance in Zickert’s thesis [Zic08] and also in [GGZ15]. Below, we follow the detailed exposition by Dunfield given in [BDRV, Sec.10.2-10.3].

A solution of the gluing equations gives rise to a developing map $\widetilde{M}\to\mathbbm{H}^{3}$ from the universal cover $\widetilde{M}$ to the 3-dimensional hyperbolic space $\mathbbm{H}^{3}$. Since the orientation preserving isometries of $\mathbbm{H}^{3}$ are in $\mathrm{PSL}(2,\mathbbm{C})$, this in turn gives rise to a $\mathrm{PSL}(2,\mathbbm{C})$ representation of the fundamental group $\pi_{1}(M)$, well-defined up to conjugation. What’s more, we get a decorated representation (those were called augmented representations in Dunfield’s terminology). Following the notation of [BDRV, Sec.10.2-10.3], let $\overline{X}(M,\mathrm{PSL}(2,\mathbbm{C}))$ denote the augmented character variety of $M$. Thus, we get a map:

[TABLE]

So far, $M$ can have boundary components of arbitrary genus. When the boundary $\partial M$ consists of a single torus boundary component, and $\gamma$ is a simple closed curve on $\partial M$, the holonomy of an augmented representation gives a regular function $h_{\gamma}:\overline{X}(M,\mathrm{PSL}(2,\mathbbm{C}))\to\mathbbm{C}^{*}$. Note that for a decorated representation $\rho$, the set of squares of the eigenvalues of $\rho(\gamma)\in\mathrm{PSL}(2,\mathbbm{C})$ is given by $\{h_{\gamma}(\rho),h_{\gamma}(\rho)^{-1}\}$. Once we fix a pair of meridian and longitude $(\mu,\lambda)$ of the boundary torus, then we get a map

[TABLE]

The defining polynomial of the 1-dimensional components of the above map is the $A$-polynomial of the 3-manifold $M$. Technically, this is the $\mathrm{PSL}(2,\mathbbm{C})$-version of the $A$-polynomial and its precise relation with the $\mathrm{SL}(2,\mathbbm{C})$-version of the $A$-polynomial (as defined by [CCG*+*94]) is discussed in detail in Champanerkar’s thesis [Cha03]; see also [BDRV, Sec.10.2-10.3].

We should point out that although (9) is a map of affine varieties, its image may miss components of $\overline{X}(M,\mathrm{PSL}(2,\mathbbm{C}))$, and hence the gluing equations of the triangulation may not detect some factors of the $A$-polynomial. In fact, when the boundary of $M$ consists of tori, the image of (9) always misses the components of abelian $\mathrm{SL}(2,\mathbbm{C})$ representations (and every knot complement has a canonical such component), but it may also miss others. For instance, there is a 5-tetrahedron ideal triangulation of the $4_{1}$ knot with an edge of valency one, and for that triangulation, $\mathcal{G}_{\mathcal{T}}$ is empty.

For later use, let us record how to compute the holonomy of a peripheral curve on the gluing equations variety. Given a path $\gamma$ in a component of $\partial M$ that is normal with respect to this triangulation, it intersects the triangles of $\partial M$ in segment joining different sides. Each segment may go from one side of the triangle to either the adjacent left side or right side. Also it separates one corner of the triangle from the other two; this corner correspond to a shape parameter which we name $z_{left}$ or $z_{right}$ depending whether the segment goes left or right. The holonomy of $\gamma$ is then:

[TABLE]

2.2. Spines and gluing equations

The ideal triangulations that we that we will discuss in the next section come from a planar projection of a knot, and it will be easier to work with their spines, that is the the dual $2$-skeleton. Because of this reason, we discuss the gluing equations of an ideal triangulation $\mathcal{T}$ in terms of its spine. In that case, edges of $\mathcal{T}$ are dual to 2-cells of the spine, and give rise to gluing equations. Recall that a spine $S$ of $M$ is a CW-complex embedded in $M$, such that each point of $S$ has a neighborhood homeomorphic to either $D^{2}$, $Y\times[0,1]$ where $Y$ is the $Y$-shaped graph or to the cone over the edges of a tetrahedron, and such that $M{\smallsetminus}S$ is homeomorphic to $\partial M\times[0,1)$. Points of the third type are vertices of the spine, points of the second type form the edges of the spines and points of the first type form the regions of the spine.

For any ideal triangulation of $M$, the dual spine is obtained as shown in Figure 1. Shape parameters that were assigned to tetrahedra are now assigned to vertices of the spine. At each vertex, two opposite corners bear the same shape parameter $z$, and the other bear the parameters $z^{\prime},z^{\prime\prime}$ according to the cyclic ordering (see Figure 1). Edge equations translate into region equations, the region equation associated to the region $R$ being:

[TABLE]

For a path $\gamma$ on the spine $S$ that is in normal position with respects to $S$, it intersects each region in a collection of segments $(\gamma_{i})_{i\in I}$. The holonomy of the segment $\gamma_{i}$ is

[TABLE]

where left and right corners are defined as in Figure 2, and the holonomy of $\gamma$ is

[TABLE]

2.3. The octahedral decomposition of a knot diagram

In this section we fix a diagram $D$ in $S^{2}$ of an oriented knot $K$. By diagram, we mean an embedded 4-valent graph in the plane, with an overcrossing/undercrossing choice at each vertex. Let $X(D)$ and $c(D)$ denote the set and the number of crossings of $D$. In this section as well as the remainder of the paper, an arc of $D$ will be the segment of the diagram joining two successive crossings of $D$. An overpass (resp. underpass) will be a small portion of the upper strand (resp. lower strand) of a crossing. We will denote the set of overpasses by $O(D)$ and the set of underpasses by $U(D)$. An overarc (resp. underarc) will be the portion of the knot joining two successive underpasses (resp. overpasses). An overarc of $K$ may pass through some number of crossings of $K$, doing so as the upper strand each time.

Given a diagram $D$ of the knot $K$ with $c(D)$ crossings, let $B_{1}$ be some ball lying above the projection plane and $B_{2}$ another ball lying under the projection plane. A classical construction, first introduced by Weeks in his thesis, and implemented in SnapPy as a method of constructing ideal triangulations of planar projections of knots [CDW, Wee05], yields a decomposition of $S^{3}{\smallsetminus}(K\cup B_{1}\cup B_{2})$ into $c(D)$ ideal octahedra. The decomposition works as follows: at each crossing of $K$, put an octahedron whose top vertex is on the overpass and bottom vertex is on the underpass. Pull the two middle vertices lying on the two sides of the overpass up towards $B_{1}$ and the two other middle vertices down towards $B_{2}$. One can then patch all these octahedra together to get a decomposition of $S^{3}{\smallsetminus}(K\cup B_{1}\cup B_{2})$. We refer to [KKY18] as well as [Thu99] for figures and more details on this construction.

From the octahedral decomposition of $S^{3}{\smallsetminus}(K\cup B_{1}\cup B_{2})$, one can get an ideal triangulation of $S^{3}{\smallsetminus}(K\cup B_{1}\cup B_{2})$ simply by splitting the octahedra further into tetrahedra. There are two natural possibilities for this splitting, as one can cut each octahedra into either $4$ or $5$ tetrahedra as shown in Figure 3. We will be interested in the decomposition where we split each octahedra into $5$ tetrahedra, obtaining thus a decomposition of $S^{3}{\smallsetminus}(K\cup B_{1}\cup B_{2})$ into $5c(D)$ tetrahedra. We denote this ideal triangulation by $\mathcal{T}^{5T}_{D}$, and we call it the “$5T$-triangulation of $D$”.

Since the inclusion map $S^{3}{\smallsetminus}(K\cup B_{1}\cup B_{2})\to S^{3}{\smallsetminus}K$ is an isomorphism on fundamental groups, a solution to the gluing equations of $\mathcal{T}^{5T}_{D}$ gives rise to a decorated $\mathrm{PSL}(2,\mathbbm{C})$ representation of the knot complement.

2.4. The spine of the $5T$-triangulation of a knot diagram

and its gluing equations

Let $\mathcal{G}_{D}$ denote the gluing equation variety of $\mathcal{T}^{5T}_{D}$. To write down the equations of $\mathcal{G}_{D}$, we will work with the dual spine, and use the spine formulation of the gluing equations introduced in Section 2.1. We describe this spine just below. This well-known spine is studied in detail by several authors including [KKY18].

Figure 4 shows a picture of the spine near a crossing of $D$. The spine contains $5$ vertices near each crossing of $D$ and can be described as follows:

First we embed $K$ in $S^{3}$ as a solid torus sitting in the middle of the projection plane; except for overpasses which go above the projection plane and underpasses which go below. We let the boundary of a tubular neighborhood of $K$ to be a subset of the spine. At each crossing we connect the overpass and the underpass using two triangles that intersects transversally in one point. Finally we glue the regions of the projection plane that lie outside $D$ to the rest of the spine. The regions of the spine are then of $3$ types:

• •

An upper/lower triangle region for each crossing, and $2c(D)$ in total.

• •

For each region of $D$ one gets an horizontal region in the spine; we call these big regions, $c(D)+2$ in total.

• •

The boundary of a neighborhood of $K$ is cut by the triangle regions and the big regions into regions lying over the projection plane (upper shingle region) and some lying under the projection plane (lower shingle regions). Note that upper shingle regions start and end at underpasses; they are in correspondance with the overarcs of the diagram, $c(D)$ in total. Similarly, the lower shingle regions are in correspondance with underarcs, and there is also $c(D)$ of them.

We now assign shape parameters to each vertex of the spine as shown in Figure 4. There are $5$ shape parameters for each crossing $c$: a central one which we call $w_{c}$ and $4$ others: $z_{c,li},z_{c,lo},z_{c,ui},z_{c,uo}$ standing for lower-in, lower-out, upper-in and upper-out. When the crossing $c$ we consider is clear, we will sometimes write $w,z_{li},z_{lo}\ldots$ dropping the index $c$.

Note that the assignment of shape parameters is such that the main version of the parameter $w,z_{li},\ldots$ lies on a corner of a triangle region, while the auxiliary $w^{\prime},w^{\prime\prime},z_{li}^{\prime},z_{li}^{\prime\prime}\ldots$ are prescribed by the cyclic ordering induced by the boundary of $S^{3}{\smallsetminus}(K\cup B_{1}\cup B_{2})$.

We can now write down the gluing equations coming from the $5T$-spine:

$\bullet$ The upper/lower triangle equations are (in the notation of Figure 4)

[TABLE]

$\bullet$ The upper/lower shingle equations. Consider an upper shingle region corresponding to an overarc going from some crossing labelled $1$ to the crossing $n$, going through crossings $1,2,\ldots,n-1$ as overpasses. Then the shingle region has one corner for each of its ends, and $4$ corners for each overpasses, as explained in Figure 5. We get:

[TABLE]

Lemma 2.1**.**

The upper/lower shingle equations have the equivalent forms, respectively:

[TABLE]

[TABLE]

Proof.

Grouping together shape parameters coming from the same vertex and using $zz^{\prime}z^{\prime\prime}=-1$, we get:

[TABLE]

and then, using Equation (11):

[TABLE]

Finally, using Equation (11), we can rewrite this as equation (12) between only $z_{lo}$’s (or only $z_{li}$’s) parameters.

Similarly for a lower shingle region corresponding to an underarc running from crossing $1$ to crossing $n$, one gets an equation:

[TABLE]

which simplifies to (13). ∎

$\bullet$ Figure 6 shows a top-view of the $5T$-spine near a crossing, as well as the shape parameters of horizontal corners of the spine. We see that each vertex of a region of $K$ gives rise to $3$ corners in the corresponding big region. For each region $R_{i}$ of $K$, we get a big region equation of the form

[TABLE]

where the corner factors $f(v)$ are prescribed by the rule shown in Figure 6.

Below, we will denote the triangle, region and shingle equations by $t_{i}$, $r_{k}$ and $s_{j}$ respectively. The above discussion defines the gluing equations variety $\mathcal{G}_{D}$ as an affine subvariety of $({\mathbb{C}}^{**})^{5c(D)}$ defined by

[TABLE]

We now express the holonomies $w_{\mu}=h_{\mu}$ and $w_{\lambda}=h_{\lambda}$ of the meridian $\mu$ and zero winding number longitude $\lambda$ in terms of the above shape parameters. Note that if $K$ is not the unknot, it is always possible to find in the diagram of $K$ an underpass that is followed by an overpass that corresponds to a different crossing of $K$. We then name those two crossings $1$ and $2$. Assume that the meridian is positioned as shown in Figure 7. Then the rule described in Section 2.1 gives us the following holonomy:

[TABLE]

As $z_{2,ui}z_{2,ui}^{\prime}z_{2,ui}^{\prime\prime}=-1$, we get:

[TABLE]

Finally, we turn to the holonomy of a longitude. We first compute the holonomy of the longitude $\tilde{l}$ corresponding to the blackboard framing of the knot. We can represent this longitude on the diagram $D$ as a right parallel of $D$. We draw this longitude on the spine in Figure 8, we can see that it intersects each upper or lower shingle region in one segment.

We compute the holonomy of each segment in an upper shingle using the convention

[TABLE]

and each lower shingle segment using the convention

[TABLE]

We can actually ignore the $-1$ signs as there are $2c(D)$ segments, an even number.

As Figure 8 shows, we get:

[TABLE]

The last product is over the set $X(D)$ of crossings of $D$, and for simplicity we do not indicate the dependence of the variables on the crossing $c\in X(D)$. Let $\lambda$ be the longitude with zero winding number with $K$. The winding number of the blackboard framing longitude $\tilde{\lambda}$ is the writhe $\mathrm{wr}(D)$ of the diagram $D$, which can be computed by $\mathrm{wr}(D)=c_{+}-c_{-}$, where $c_{+}$ and $c_{-}$ are the number of positive and negative crossings of the diagram. We then have $\tilde{\lambda}=\lambda\mu^{wr(D)}$ and thus

[TABLE]

2.5. Labeled knot diagrams

In this section we introduce a labeling of the crossings in a knot diagram, closely related to the Dowker-Thistlethwaite notation of knots.

Recall that $D$ is a planar diagram of an oriented knot $K$ and that we have chosen two special crossings $1$ and $2$ that are successive in the diagram, such that such crossing $1$ corresponds to an underpass and crossing $2$ to an overpass. This choice determines a labeling of crossings of $D$ as follows.

Following the knot, we label the other crossings $3,4,\ldots.$ Note that as the knot passes through each crossing twice, each crossing $c$ of $D$ gets two labels $j<j^{\prime}$. Exactly one of those two labels correspond to the overpass and the other one to the underpass. Arcs of the diagram join two successive over- or underpasses labeled $l$ and $l+1$ (or $2c(D)$ and $1$). We write $[l,l+1]$ for the arc joining crossings $l$ and $l+1$.

This labeling is illustrated in Figure 9 in the case of the Figure eight knot.

2.6. Analysis of triangle and shingle relations

In this section, we show that the triangle and shingle equations allow us to eliminate variables in the gluing variety $\mathcal{G}_{D}$. We have the following:

Proposition 2.2**.**

In $\mathcal{G}_{D}$, each of the variables $w_{c},z_{c,li},z_{c,lo},z_{c,ui},z_{c,uo}$ are monomials in the variables $w_{c},w_{\mu}$ and $w_{0}=z_{1,lo}$.

Proof.

Fix a labeled knot diagram $D$ as in Section 2.5. Before eliminating variables, we start by assigning to each arc $[l,l+1]$ of the diagram a new parameter $z_{l,l+1}$. These parameters are expressed in terms of the previous parameters by the following rules:

[TABLE]

We recall that in the above $O(D)$ (resp. $U(D)$) is the set of overpasses (resp. underpasses) in the diagram $D$. Also, given integers $a,b\in{\mathbb{Z}}$ with $a\leq b$, we denote

[TABLE]

Note that the arc parameters $z_{l,l+1}$ are all clearly monomials in $w_{0}$ and the $w_{c}$’s.

We claim that each of the shape parameters $z_{c,li},z_{c,lo},z_{c,ui},z_{c,uo}$ are monomials in the $z_{l,l+1}$’s and $w_{\mu}$. This will imply the proposition. Indeed, let $[k,k+1]$ be an arc of $K$. Then we claim that:

[TABLE]

Note $z_{1,2}=z_{1,lo}$ by definition. If $k$ is an underpass, the formula

[TABLE]

matches with the upper shingle equation expressing $z_{k,lo}$ in terms of $z_{1,lo}$. Indeed, if $k$ is the underpass coming immediately after underpass $1$, Equation (12) says:

[TABLE]

As crossings $2,3,\ldots k-1$ correspond to overpasses and $k$ to an underpass, we also have

[TABLE]

By induction, we find that $z_{k,k+1}=z_{k,lo}$ for any underpass $k$.

The second case is then a consequence of the lower triangle equation $z_{k+1,li}=\frac{1}{w_{k+1}z_{k+1,lo}}$, and the fact that $z_{k,k+1}=z_{k+1,k+2}w_{k+1}$ as $k+1$ is an underpass.

Note that $z_{2,ui}=\frac{z_{1,lo}}{w_{\mu}}$ by Equation (16), so the fourth case is valid for the arc $[1,2]$. Similarly to case 1, we can prove case 4 for other arcs ending in an overpass from the lower shingle equations by induction.

Finally, the third case follows as $z_{k,uo}=\frac{1}{w_{k}z_{k,ui}}$, and $z_{k,k+1}=w_{k}z_{k-1,k}$. ∎

In the rest of the paper, we will often use the arc parameters $z_{k,k+1}$ defined above to express equations in $\mathcal{G}_{D}.$

For instance, thanks to Proposition 2.2, we can rewrite the big region equations $r_{k}=1$ as equations $r_{k}(w)=1$, where $r_{k}(w)$ is expressed in terms of the variables $w$ only.

Remark 2.3**.**

Although the arc parameters $z_{l,l+1}$ are just monomials in the $w$ variables, they are helpful for writing down the equations defining $\mathcal{G}_{D}$ in a more compact way. When the choice of a crossing $c$ is implicit, we introduce a simplified notation for the parameters associated to arcs neighboring $c$. We will write $z_{a},z_{b},z_{a^{\prime}},z_{b^{\prime}}$ for the parameters associated to the inward half of the overpass, inward half of underpass, outward half of underpass and outward half of underpass.

With this convention, at any crossing we have:

[TABLE]

For instance, we get a new expression of the holonomy of the longitude:

Proposition 2.4**.**

With the convention of Remark 2.3, the holonomy of the zero-winding number longitude is expressed by:

[TABLE]

Proof.

By Equation (17) we have:

[TABLE]

∎

2.7. Analysis of big region equations

Recall that the big region equations are parametrized by the regions of the planar diagram $D$, i.e., by the connected components of $S^{2}{\smallsetminus}D$. In this section, we give an alternative set of equations which are parametrized by the crossings of $D$, and we call those the loop equations.

Our motivation comes from the fact that we will later match the loop equations with equations that come from a state sum formula for the colored Jones polynomial.

Consider a crossing $c$ in the labeled diagram $D$. Recall from Section 2.5 that $c$ has two labels $j<j^{\prime}$. The arc $[j,j^{\prime}]$ starts and ends at the same crossing, hence one may close it up to obtain a loop $\gamma_{c}$. For a region $R_{i}$ of the diagram, let us pick a point $p_{i}$ in the interior of $R_{i}$. We write $w(\gamma_{c},p_{i})$ for the winding number of $\gamma$ relative to the point $p_{i}$. The big region equation corresponding to the region $R_{i}$ is $r_{i}=1$, where $r_{i}$ is the product of corners factors, see Equation (14) and Figure 6. The loop equation $L_{c}=1$ is then defined by

[TABLE]

We also introduce

[TABLE]

Proposition 2.5**.**

The set of equations $L_{0}=1$, $L_{c}=1$ for all $c\in X(D)$ is equivalent to the set of equations $r_{i}=1$ for all region $R_{i}$ of $D$.

Proof.

The equations $L_{0}=1$, $L_{c}=1$ are clearly implied by the big region equations $r_{i}=1$ as the $L_{c}$’s and $L_{0}$ are monomials in the $r_{i}$’s. We will show that the $r_{i}$’s are also monomials in $L_{0}$ and the $L_{c}$’s, and thus equations $r_{i}=1$ are a consequence of loop equations.

Let us consider the diagram $D$ as an oriented $4$-valent graph embedded in $S^{2}$. For any $\delta\in H_{1}(D,{\mathbb{Z}})$, we can also introduce a loop equation

[TABLE]

Note that $\delta\rightarrow L_{\delta}$ is a morphism of group $H_{1}(D,{\mathbb{Z}})\rightarrow{\mathbb{C}}^{*}$ and that the equation $r_{i}$ can be presented in this form too:

Indeed, chose $\delta=\partial R_{i}$ with positive orientation. Then $w(\delta,p_{j})=0$ if $j\neq i$, and $w(\delta,p_{i})=1$, hence $L_{\delta}=r_{i}$.

Thus we only need to prove that $H_{1}(D,{\mathbb{Z}})$ is generated by $K$ and the classes $\gamma_{c}$. The diagram $D$ has $c(D)$ vertices and $2c(D)$ edges, and thus $H_{1}(D,{\mathbb{Z}})={\mathbb{Z}}^{c(D)+1}$. So we need to show that $K$ and the loops $\gamma_{c}$ are a ${\mathbb{Z}}$-basis of $H_{1}(D,{\mathbb{Z}})$. To do this we first show that they are linearly independent in the space of $1$-chains $C_{1}(D,{\mathbb{Z}})$.

Recall that we fixed a labeling of overpasses and underpasses in $[1,2c(D)]$ following the knot $K$. Note that the arcs $[1,2],[2,3],\ldots[2c(D),1]$ give a basis of $C_{1}(D,{\mathbb{Z}})$. We order this basis with the convention $[1,2]<[2,3]<\ldots<[2c(D),1]$.

Then $K=[1,2]+[2,3]+\ldots+[2c(D),1]$ in $C_{1}(D,{\mathbb{Z}})$, and if a crossing $c$ has labels $j<j^{\prime}$, then $\gamma_{c}=[j,j+1]+\ldots+[j^{\prime}-1,j^{\prime}]$.

We see that $K$ is not in the space generated by the $\gamma_{c}$ as it is the only one with non-zero coordinate along $[2c(D),1]$.

Moreover, the loops $\gamma_{c}$ are linearly independent as the indices of their first non-zero coordinates are all different.

So $K$ and the $\gamma_{c}$ are linearly independent in $H_{1}(D,{\mathbb{Z}})$, and thus a ${\mathbb{Q}}$-basis of $H_{1}(D,{\mathbb{Q}})$. We can actually show that they form a ${\mathbb{Z}}$-basis of $H_{1}(D,{\mathbb{Z}})$. Indeed if $\delta\in H_{1}(D,{\mathbb{Z}})$, we can subtract a ${\mathbb{Z}}$-linear combination of $K$ and the $\gamma_{c}$’s to $\delta$ to obtain an element with [math] coordinate on $[2c(D),1]$ and each $[j,j+1]$ for each crossing with labels $j<j^{\prime}$. This element has then to be zero as $(K,\gamma_{c})$ is a ${\mathbb{Q}}$-basis of $H_{1}(D,{\mathbb{Q}})$.

Thus $K$ and the $\gamma_{c}$’s generate $H_{1}(D,{\mathbb{Z}})$, and the $r_{i}$’s are monomials in the $L_{0},L_{c}$. ∎

2.8. Formulas for the loop equations

In this section, we simplify the equations $L_{0},L_{c}$ which we defined as monomials in the big region equations. Our goal is to express those equations in terms of the arc parameters $z_{k,k+1}$ introduced in Section 2.6, which we recall are monomials in the $w$ variables.

Proposition 2.6**.**

Let $c$ be a crossing of $D$ with labels $j<j^{\prime}$. For $k\in[j,j^{\prime}]$, let $\varepsilon(k)=1$ if $k$ corresponds to a positive crossing and $\varepsilon(k)=-1$ otherwise. Let also $u_{+}(k)=\frac{1+\varepsilon(k)}{2}$ and $u_{-}(k)=\frac{1-\varepsilon(k)}{2}$. Then we have:

[TABLE]

where in the above we set

[TABLE]

Proof.

We recall that $\gamma$ is the loop obtained from the arc $[j,j^{\prime}]$ of $D$ by gluing its two ends together. Let also $\gamma^{\prime}$ be the complementary loop of $\gamma$, which is obtained from the arc $[j^{\prime},j]$ by gluing the two ends. Note that $\gamma^{\prime}$ goes through the underpass labeled $1$.

As $L_{c}=\underset{R_{i}\ \textrm{region}}{\prod}r_{i}^{w(\gamma,p_{i})}$ is a product of big region equations, and each big region factor is a product of corner factors, we can rewrite $L_{c}$ as a product of corner factors. Each corner $v$ of $D$ appears in one region $R_{i}$ only , and the winding number $w(\gamma,v)$ of $\gamma$ around $v$ is the same as $w(\gamma,p_{i})$. Thus we may rewrite $L_{c}$ as

[TABLE]

where the corner factors $f(v)$ are those of Figure 6.

Figure 10 shows the local pattern of winding numbers of corners near a crossing of $D$, depending which neighboring arcs belong to $\gamma$ and $\gamma^{\prime}$. First let us note for a crossing between two strands of $\gamma^{\prime}$, all local winding numbers are equal, thus the crossing contributes by the product of all $4$ corners factors to some power. However, at any positive crossing, the product of corner factors is

[TABLE]

by the rule $zz^{\prime}z^{\prime\prime}=-1$ and the triangle equations. Similarly, at any positive crossing, the product of corner factors is

[TABLE]

So crossings between two strands of $\gamma^{\prime}$ do not contribute to $L_{c}$.

Next we consider a crossing between one strand of $\gamma$ and one strand of $\gamma^{\prime}$. By the local winding numbers shown in Figure 10 and that fact that the product of the $4$ corner factors at a crossing is $1$, such a crossing contributes by the product of the two corner factors to the left of $\gamma$. Similarly, for a crossing between two strands of $\gamma$, we get the product of the two corner factors to the left of the first strand times the two corner factors to the left of the other strand.

Hence, each overpass or underpass $l\in[\![j+1,j^{\prime}-1]\!]$ of $\gamma$ contributes to one factor $K_{l}$ which is the product of the two left corner factors. By the rule described in Figure 6, for a positive overpass we get

[TABLE]

where the last equality comes from the fact that, at any crossing, $\frac{z_{a^{\prime}}}{z_{a}}=\frac{z_{b}}{z_{b^{\prime}}}=w$. Similarly, at a negative overpass we get:

[TABLE]

At a positive underpass we get:

[TABLE]

and, finally, at a negative underpass we get:

[TABLE]

All those overpass/underpass factors correspond to the ones in Equation (21). Finally we turn to the contribution $K_{c}$ of crossing $c$. By the local pattern of winding numbers in Figure 10, and the corner factors rule of Figure 6, we have, if $j$ is a positive overpass:

[TABLE]

If $j$ is a negative overpass, we have:

[TABLE]

If $j$ is a positive underpass, then:

[TABLE]

Finally if $j$ is a negative underpass, then:

[TABLE]

We clearly see that in each case the factor $K_{c}$ matches with that of Proposition 2.6. ∎

We want to rearrange the loop equations slightly, grouping together the factors $w_{\mu}^{\varepsilon(k)}$ on the one side and the factors $\frac{z_{b}^{u_{-}(k)}}{z_{b^{\prime}}^{u_{+}(k)}}$ and $\frac{z_{a}^{u_{-}(k)}}{z_{a^{\prime}}^{u_{+}(k)}}$ on the other side. For the former we claim:

Lemma 2.7**.**

Let $c$ be a crossing of $D$ with labels $j<j^{\prime}$, $\gamma$ the loop $[j,j^{\prime}]/_{j=j^{\prime}}$, and $\gamma^{\prime}$ the loop $[j^{\prime},j]/_{j=j^{\prime}}$. For $l\in(j,j^{\prime})$ an over- or underpass, let $\varepsilon(l)$ be the sign of the corresponding crossing. Then we have

[TABLE]

Remark 2.8**.**

By the above lemma, the factors $w_{\mu}^{\varepsilon(k)}$ in the product on the right of Equation (21) group up to one factor $w_{\mu}^{\mathrm{wr}(\gamma)+\mathrm{lk}(\gamma,\gamma^{\prime})}$.

Proof.

The crossings of $D$ that are in $(j,j^{\prime})$ are of two types: self-crossings of $\gamma$ and crossing between $\gamma$ and $\gamma^{\prime}$. Self-crossings of $\gamma$ belong to both an overpass and an underpass $l\in(j,j^{\prime})$, hence in both sums in the lemma, those crossings contribute to $c_{+}(\gamma)-c_{-}(\gamma)=\mathrm{wr}(\gamma)$.

Moreover the linking number of $\gamma$ and $\gamma^{\prime}$ can be computed in two ways as $\underset{l\in\gamma\cap\gamma^{\prime}}{\sum}\frac{\varepsilon(l)}{2}$ or as $\underset{l\in\gamma\cap\gamma^{\prime}\cap U(D)}{\sum}\varepsilon(l)$. Thus hence in both sums in the lemma mixed crossings contribute to $\mathrm{lk}(\gamma,\gamma^{\prime})$. ∎

Lemma 2.9**.**

Let $c$ be a crossing of $D$ with labels $j<j^{\prime}$. Then:

[TABLE]

where $C=\left(\frac{z_{b_{c}}}{z_{a_{c}^{\prime}}}\right)^{\frac{1}{2}}=\left(\frac{z_{b_{c}^{\prime}}}{z_{a_{c}}}\right)^{\frac{1}{2}}$ if $j$ is an overpass and $C=\left(\frac{z_{a_{c}}}{z_{b_{c}^{\prime}}}\right)^{\frac{1}{2}}=\left(\frac{z_{a_{c}^{\prime}}}{z_{b_{c}}}\right)^{\frac{1}{2}}$.

Proof.

We have by definition of $u_{+}(k)$ and $u_{-}(k)$:

[TABLE]

Moreover, as at any crossing $\frac{z_{b}}{z_{b^{\prime}}}=\frac{z_{a^{\prime}}}{z_{a}}=w$, we have:

[TABLE]

Finally, if $j$ is an overpass then $\left(\frac{z_{j^{\prime}-1,j^{\prime}}}{z_{j,j+1}}\right)^{\frac{1}{2}}=\left(\frac{z_{b_{c}}}{z_{a_{c}^{\prime}}}\right)^{\frac{1}{2}}$ as $z_{a_{c}^{\prime}}=z_{j,j+1}$ and $z_{b_{c}}=z_{j^{\prime}-1,j^{\prime}}$. Similarly, $\left(\frac{z_{j^{\prime}-1,j^{\prime}}}{z_{j,j+1}}\right)^{\frac{1}{2}}=\left(\frac{z_{a_{c}}}{z_{b_{c}^{\prime}}}\right)^{\frac{1}{2}}$ if $j$ is an underpass. ∎

From Proposition 2.6 together with Lemma 2.7 and 2.9, we obtain another formula for the loop equation:

Proposition 2.10**.**

Let $c$ be a crossing of $D$ with labels $j<j^{\prime}$ and let $L_{c}$ be the associated loop equation. If $k\in(j,j^{\prime})$, let $\varepsilon(k)$ be the sign of the corresponding crossing. Then:

[TABLE]

where $K_{c}^{\prime}$ is obtained from $K_{c}$ of Proposition 2.6 by replacing respectively a factor $\left(\frac{1}{z_{b_{c}^{\prime}}}\right)$, $z_{a_{c}^{\prime}}$, $\left(\frac{1}{z_{a_{c}^{\prime}}}\right)$, or $z_{b_{c}^{\prime}}$ by $\frac{1}{(z_{a_{c}}z_{b_{c}^{\prime}})^{\frac{1}{2}}}$, $(z_{a_{c}^{\prime}}z_{b_{c}})^{\frac{1}{2}}$, $\frac{1}{(z_{a_{c}^{\prime}}z_{b_{c}})^{\frac{1}{2}}}$, or $(z_{a_{c}}z_{b_{c}^{\prime}})^{\frac{1}{2}}$ if $j$ is a positive overpass, a negative overpass, a positive underpass or a negative underpass.

Finally, we turn to the expression of the last loop equation $L_{0}=\underset{R_{i}\ \textrm{region}}{\prod}r_{i}^{w(K,R_{i})}$ that we introduced in Section 2.7.

Proposition 2.11**.**

We have the formula:

[TABLE]

Proof.

We proceed similarly as in the proof of 2.6. As we are taking the whole knot $K$ instead of one of the loops $\gamma_{c}$, the local pattern of winding numbers at any crossing looks like the third drawing in Figure 10.

By the corner factor rule of Figure 6, we get a factor

[TABLE]

at a positive crossing and a factor:

[TABLE]

at a negative crossing, using that $z_{a}z_{b}=z_{a^{\prime}}z_{b^{\prime}}$ at any crossing. ∎

2.9. A square root of the holonomy of the longitude

In this section, we show that the holonomy of the longitude $w_{\lambda}$ admits a square root in ${\mathbb{C}}[\mathcal{G}_{D}]$. We prove the following.

Proposition 2.12**.**

Let $s$ be defined by

[TABLE]

Then $s\in{\mathbb{C}}(w_{\mu},w_{0},w_{c})$ and $s^{2}=\frac{1}{w_{\lambda}L_{0}}$.

Proof.

By Equation (18),

[TABLE]

and by Equation (22):

[TABLE]

Those two equations clearly imply that $s^{2}=\frac{1}{w_{\lambda}L_{0}}$. The non-trivial part is to show that $s$ is actually in ${\mathbb{C}}(w_{\mu},w_{0},w_{c})$, which is equivalent to showing the degree of the monomial $\underset{X(D)}{\prod}wz_{a}z_{b}$ is even in each of the variable $w_{\mu},w_{0}$ and $w_{c}$.

First we note that all arc parameters $z_{a},z_{b}$ have degree [math] along $w_{\mu}$ and degree $1$ along $w_{0}$. So what we need to show is that the product $\underset{X(D)}{\prod}z_{a}z_{b}$ has odd degree along each variable $w_{c}$ associated to a crossing. We remark that this product is also the product of all arc parameters as each arc is an inward arc of exactly one crossing.

Let $c$ be a crossing with labels $j<j^{\prime}$. Then for any arc $[k,k+1]$ the arc parameter $z_{k,k+1}$ is of the form $z_{k,k+1}=w_{0}w_{c}^{\varepsilon}\underset{c^{\prime}\neq c}{\prod}w_{c^{\prime}}^{\varepsilon_{c^{\prime}}}$, where $\varepsilon\in\{-1,0,1\}$, and $\varepsilon\neq 0$ if and only if $[k,k+1]\subset[j,j^{\prime}]$. So all we have to show is that $j^{\prime}-j$ is always odd for any crossing $c$. The reason is that the loop $\gamma=[j,j^{\prime}]/_{j\sim j^{\prime}}$ has $j^{\prime}-j-1$ intersection points with the rest of $K$, and those intersection points bound a collection of segments, which are the intersection of $K$ with a disk bounded by $\gamma$. So $j^{\prime}-j-1$ is always even. ∎

3. $q$-holonomic functions, creative telescoping and

certificates

In this section we recall some properties of $q$-holonomic functions, creative telescoping and certificates, which we will combine with a state sum formula for the colored Jones polynomial to prove our main Theorem 1.1. Recall that a $q$-holonomic function $f:\mathbbm{Z}\to\mathbbm{Q}(q)$ is one that satisfies a non-zero recursion relation of the form (1), i.e., a function with annihilator (3) satisfying $\mathrm{Ann}(f)\neq 0$. $q$-holonomic functions of several variables are defined using a notion of Hilbert series dimension, and are closed under sums, products as well as summation of some of their variables. Building blocks of $q$-holonomic functions are the proper $q$-hypergeometric functions of [WZ92]. For a detailed discussion of $q$-holonomic functions, we refer the reader to the survey article [GL16].

The following proposition is the fundamental theorem of $q$-holonomic functions. When $F$ is proper $q$-hypergeometric, a proof was given in Wilf-Zeilberger [WZ92]. A detailed proof of the next proposition, as well as a self-contained introduction to $q$-holonomic functions, we refer the reader to [GL16].

Proposition 3.1**.**

(a) Proper $q$-hypergeometric functions are $q$-holonomic.

(b) Let $F:\mathbbm{Z}^{r+1}\to\mathbbm{Q}(q)$ be $q$-holonomic in the variables $(n,k)\in\mathbbm{Z}\times\mathbbm{Z}^{r}$ such that $F(n,\cdot)$ has finite support for any $n$ and let $f:\mathbbm{Z}\to\mathbbm{Q}(q)$ be defined by

[TABLE]

Then $f$ is $q$-holonomic.

The above proposition combined with an $R$-matrix state-sum formula for the colored Jones polynomial implies that the colored Jones polynomial of a knot (or link, colored by representations of a fixed simple Lie algebra) is $q$-holonomic [GL05].

With the notation of the above proposition, a natural question is how to compute $\mathrm{Ann}(f)$ given $\mathrm{Ann}(F)$. This is a difficult problem practically unsolved. However, an easier question can be solved: namely given $\mathrm{Ann}(F)$, how to compute a nonzero element in $\mathrm{Ann}(f)$. The answer to this question is given by certificates, which are synonymous to the method of creative telescoping, coined by Zeilberger [Zei91]. The latter aims at computing recursions for holonomic functions obtained by summing/integrating all but one variables. For a detailed discussion and applications, see [PWZ96, WZ92] and also [BLS13].

Proposition 3.2**.**

(a) Let $F$ and $f$ be as in Proposition 3.1, and consider the map $\varphi$ from (6). Let

[TABLE]

Then $\varphi(P)\in\mathrm{Ann}(f)$.

(b) There exist $P$ as above with $\varphi(P)\neq 0$.

Nonzero elements $P$ as in (24) are called “certificates”, and those that satisfy $\varphi(P)\neq 0$ are called “good certificates”. Certificates are usually computed in the intersection $\mathrm{Ann}(F)\cap\mathbbm{Q}(q,Q)\langle E,E_{i}\rangle$, where membership reduces to a linear algebra question over the field $\mathbbm{Q}(q,Q)$ and then lifted to the ring $\mathbbm{Q}[q,Q]\langle E,E_{i}\rangle$ by clearing denominators.

Part (b) is shown in Zeilberger [Zei90] and in detail in Koutschan’s thesis [Kou09, Thm.2.7]. In the latter reference, this is called the “elimination property” of holonomic ideals. Part (a) is easy and motivates the name “creative telescoping”. Indeed, one may write

[TABLE]

A recurrence relation of this form is also called a certificate. After expanding the sum $\underset{k\in{\mathbb{Z}}^{d}}{\sum}P(E,Q,E_{i})F(n,k)=0$, the terms

[TABLE]

are telescoping sums and thus equal to [math]. Finally, note that when $F$ is proper $q$-hypergeometric, an operator $P$ as above may be found by using its monomials as unknowns and solving a system of linear equations of $PF/F$. Hence, once $P$ is found (and that is the difficult part), it is easy to check that it satisfies the relation $PF=0$, which reduces to an identity in a field of finitely many variables–hence the name “certificate”.

Part (b) follows by multiplying an element of $\mathrm{Ann}(F)$ on the left if necessary by a monomial in $Q_{i}$. We thank C. Koutschan for pointing this out to us.

4. The colored Jones polynomial of a knot

4.1. State sum formula for the colored Jones polynomial

of a knot diagram

In this section, we use a diagram $D$ of an oriented knot $K$ to give a (state sum) formula for the $n$-th colored Jones polynomial $J_{K}(n)\in{\mathbb{Z}}[q^{\pm 1}]$ of $K$. Such a formula is obtained by placing an $R$-matrix at each crossing, coloring the arcs of the diagram with integers, and contracting tensors as described for instance in Turaev’s book [Tur94]. The formula described in this section follows the conventions introduced in [GL15]; we also refer to [GL15] for all proofs.

For $n\geqslant 0$, we define the $n$-th quantum factorial by

[TABLE]

Note that quantum factorials satisfy the recurrence relation $(q)_{n+1}=(1-q^{n+1})(q)_{n}$ for any $n\geqslant 0$. As it will be helpful for us to have recurrence relations that are valid for any $n\in{\mathbb{Z}}$, we will use the following convention of quantum factorials and their inverses:

[TABLE]

With the above definition and with the notation of (2) we have:

[TABLE]

Fix a labeled diagram $D$ of an oriented knot $K$ as in Section 2.5. After possibly performing a local rotation, one can arrange $D$ so that at each crossing the two strands of $K$ are going upwards. The diagram $D$ is then composed of two types of pieces: the crossings (which can be possible or negative) and local extrema. Let $\mathrm{arc}(D)$ be the set of arcs of the diagram $D$, we say that a coloring

[TABLE]

is $n$-admissible if the color of any arc is in $[0,n]$ and for any crossing, if $a,a^{\prime},b,b^{\prime}$ are the color of the neighboring arcs in shown in Figure 11, then $a^{\prime}-a=b-b^{\prime}=k\geqslant 0$. Let $S_{D,n}$ be the set of all $n$-admissible colorings of the arcs of $D$. Note that $S_{D,n}$ coincides with the set of lattice points in the $n$-th dilatation of a rational convex polytope $P_{D}$ defined by the $n$-admissibility conditions.

For a proof of the next proposition, we refer to [GL15, Sec.2].

Proposition 4.1**.**

The normalized $n$-th colored Jones polynomial of $K$ is obtained by the formula:

[TABLE]

where $\widetilde{w}_{D}(n,r)=\underset{X\ \textrm{piece}}{\prod}w(X,r)$ is a product of weights associated to crossings and extrema of $D$ as shown in Figure 11.

The insertion of the factor $q^{n/2}$ in front of the above sum is done for convenience only, so that $J_{K}(n)$ is a Laurent polynomial in $q$ rather than one in $q^{1/2}$. This normalization plays no role in the AJ Conjecture. Note that we have $J_{K}(0)=1$ for every knot $K$ and $J_{\mathrm{Unknot}}(n)=\frac{1-q^{n+1}}{1-q}$ for any $n\geqslant 0$ and $J_{K}(1,q^{-1})/J_{\mathrm{Unknot}}(1,q^{-1})$ is the Jones polynomial of $K$.

Note that the color of all arcs are completely determined by the shifts $(k_{1},\dots,k_{c(D)})\in{\mathbb{Z}}^{c(D)}$ associated to crossings and the color $k_{0}$ of the arc $[1,2]$. In other words, $r=r(k)$ is a linear function of $k=(k_{0},\dots,k_{c(D)})\in{\mathbb{Z}}^{c(D)+1}$. Suppressing the dependence on $q$, we abbreviate $\widetilde{w}_{D}(n,r(k))(q)$ simply by $w_{D}(n,k)$.

When examining recurrence relations for the colored Jones it will be more convenient to express $J_{K}(n)$ as a sum over all $k\in{\mathbb{Z}}^{c(D)+1}$ rather than a sum over colorings $r$ in the set $S_{D,n}$ of lattice points in the rational convex polytope $P_{D}$. For this we have the lemma:

Lemma 4.2**.**

For any knot $K$, we have:

[TABLE]

Proof.

We recall that we have set the convention $\frac{1}{(q)_{n}}=0$ if $n<0$. From the definition of weights associated to crossings, we see that at any crossing the weight vanishes unless $k\geqslant 0$, $b^{\prime}\geqslant 0$ and $a^{\prime}\leqslant n$.

Pick a coloring so that the associated weight is non-zero. Consider the color $c_{i,i+1}$ of the arc $[i,i+1]$. If $i$ is an underpass, then we get that $c_{i,i+1}\geqslant 0$. If on the other hand $i$ is an overpass, then $c_{i,i+1}=c_{i-1,i}+k_{i}$, so $c_{i,i+1}\geqslant c_{i-1,i}$. If $i-1$ is an underpass, one concludes that $c_{i,i+1}\geqslant c_{i-1,i}\geqslant 0$, else, one can continue until we meet an underpass $k$, and write

[TABLE]

Thus if the weight is non-zero, the color of all arcs must be non-negative.

Similarly, we can show that the color of all arcs muss be at most $n$. We already know that $c_{i,i+1}\leqslant n$ if $i$ is an overpass. Else, if $k$ is the overpass immediately before $i$, we have

[TABLE]

Thus any non-zero weight corresponds to an element of $S_{D,n}$. ∎

4.2. The annihilator ideal of the summand of the

state sum

It is easy to see that the summand $w_{D}(n,k)$ of the state sum (25) is a $q$-proper hypergeometric function in the sense of [WZ92]. In this section we compute generators of its annihilator ideal. To do so, we compute the effect of the shift operators $E$, $E_{0}$ and $E_{c}$ on $w_{D}(n,k)$. Each operator is acting on exactly one of the $c(D)+2$ variables $(n,k)$ leaving all others fixed.

• •

$E$ shifts $n$ to $n+1$.

• •

$E_{0}$ shifts $k_{0}$ to $k_{0}+1$. As the color of any other arc of $D$ is of the form $k_{0}+\underset{c\in X(D)}{\sum}\varepsilon_{c}k_{c}$ with $\varepsilon_{c}\in\{-1,0,1\}$, the operator $E_{0}$ actually shifts the color of all arcs up by $1$.

• •

$E_{c}$ for each crossing $c$ shifts $k_{c}$ to $k_{c}+1$.

The propositions of this section will match, after setting $q=1$, with the gluing equations of the $5T$-spine of the knot projection.

Because we will later reduce our equations by plugging $q=1$, it will only matter to us that they are exact up to fixed powers of $q$. We will write $q^{*}$ for a power of $q$ which does not depend on $(n,k)$.

Let us start by considering the effect of $E_{0}$ on $w$.

Proposition 4.3**.**

The summand $w_{D}(n,k)$ of the colored Jones polynomial satisfies:

[TABLE]

Remark 4.4**.**

The denominators in the above equations actually vanish if $k\notin S_{D,n}$. To obtain recurrence relations that are valid for any $(n,k)$, we can simply move each denominator to the other side of the equation. The convention $\frac{1}{(q)_{i}}=0$ if $i<0$ will ensure that the equations still hold.

Proof.

Let us note first that the weights of local extrema are linear powers of $q$. When computing the ratio $\frac{E_{0}w_{D}(n,k)}{w_{D}(n,k)}$ those weights will only contribute to a $q^{*}$ factor. Thus we can discard those weights while trying to prove Proposition 4.3. We can also discard any linear power $q$ from the weights of crossing for the same reason.

We also note that one can separate the weights $w(c)$ of crossings into a product of two factors $w_{>}(c)$ and $w_{<}(c)$, where

[TABLE]

and

[TABLE]

where $a,a^{\prime},b,b^{\prime}$ are the colors of arcs neighboring the crossing $c$, following the convention described in Figure 11.

Recall that $E_{0}$ shifts the color of all arcs up by $1$. Up to $q^{*}$, the ratio $\frac{E_{0}w_{D}(n,k)}{w_{D}(n,k)}$ is a product of factors $\mu(c)=\frac{E_{0}w_{>}(c)}{w_{>}(c)}$ and $\nu(c)=\frac{E_{0}w_{<}(c)}{w_{<}(c)}$ for every crossing. We compute that:

[TABLE]

and

[TABLE]

if $c$ is positive and

[TABLE]

if $c$ is negative. This gives Equation (27). ∎

Let us now turn to the effect of operator $E$.

Proposition 4.5**.**

The summand $w_{D}(n,k)$ of the colored Jones state sum satisfies:

[TABLE]

Proof.

Again, we can safely ignore the contribution of weights of local extrema and any linear power of $q$ in the weights of crossings as they just contribute to a $q^{*}$ factor. First, note that the effect of $E$ is to shift $n$ up by $1$ and leave the colors of all arcs invariant. Then, as in the previous Proposition, any crossing $c$ contributes to the ratio by the product of two factors $\mu(c)$ and $\nu(c)$, where

[TABLE]

and

[TABLE]

as $b^{\prime}=b-k$, if $c$ is a positive crossing. For $c$ a negative crossing, we have:

[TABLE]

as $a^{\prime}=a+k$. Combining the factors $\mu(c)$ and $\nu(c)$ we get Equation (28). ∎

Proposition 4.6**.**

Fix a labeled diagram $D$ as in Section 2.5. Let $c$ be a crossing of $D$ with labels $j<j^{\prime}$. Then the summand $w_{D}(n,k)$ of the colored Jones polynomial satisfies:

[TABLE]

if $j$ is an overpass and

[TABLE]

if $j$ is an underpass. In the above, we set

[TABLE]

Proof.

Let $c$ be a crossing with labels $j<j^{\prime}$. The effect of $E_{c}$ is to shift $k_{c}$ up by $1$. Note that the colors of arcs $[k,k+1]\subset[1,j]\cup[j^{\prime},1]$ do not depend on $k_{c}$, while the colors of arcs $[k,k+1]\subset[j,j^{\prime}]$ are of the form $c_{0}+\varepsilon k_{c}$, where $c_{0}$ does not depend on $k_{c}$ and $\varepsilon=1$ if $j$ is an overpass, $\varepsilon=-1$ else. Thus the effect of $E_{c}$ is to shift the colors of arcs in $[j,j^{\prime}]$ up by $1$ (if $j$ is an overpass) or down by $1$ (if $j$ is an underpass).

As before we neglect the weights of local extrema and any linear power $q$ in the weights of crossings. Let us write $a,a^{\prime},b,b^{\prime}$ for the colors of the arcs neighboring a crossing $c^{\prime}\in(j,j^{\prime})$ with labels $l<l^{\prime}$, let $k=a^{\prime}-a=b-b^{\prime}$.

First we note that the weights $w_{>}(c^{\prime})=\frac{(q)_{n-a}}{(q)_{n-a^{\prime}}}\frac{(q)_{b}}{(q)_{b^{\prime}}(q)_{k}}$ can separated into a factor $w_{>}(l)=\frac{(q)_{n-a}}{(q)_{n-a^{\prime}}}$ associated to the overpass $l$ and a factor $w_{>}(l^{\prime})=\frac{(q)_{b}}{(q)_{b^{\prime}}(q)_{k}}$ associated to the underpass $l^{\prime}$. The weights $w_{<}(c^{\prime})$ are not separable in the same way; however the ratios $\nu(c^{\prime})=\frac{E_{c}w_{<}(c^{\prime})}{w_{<}(c^{\prime})}$ are linear powers of $q$ and thus we can compute those factors up to $q^{*}$ as a product of two factors $\nu(l),\nu(l^{\prime})$, where in $\nu(l)$ we apply the shift only to the colors $a,a^{\prime}$ and in $\nu(l^{\prime})$ we apply the shift only to the colors $b,b^{\prime}$.

Now we compute the factors $\mu(l)=\frac{E_{c}w_{>}(l)}{w_{>}(l)}$ and $\nu(l)$ associated to over- or underpasses.

Note that if $l\notin[\![j,j^{\prime}]\!]$, then no arc of the over- or underpass $l$ has its color changed under the shift $E_{c}$. Thus $\mu(l),\nu(l)=1$ in this case.

Consider $l\in(j,j^{\prime})$ that corresponds to a positive crossing. Assume first that $l$ is an overpass. If $j$ is an overpass, the operator $E_{c}$ shifts the colors $a,a^{\prime}$ up by $1$, and we have

[TABLE]

If $j$ was an underpass instead, colors $a,a^{\prime}$ are shifted down by $1$ under $E_{c}$, so that

[TABLE]

Now if $l\in(j,j^{\prime})$ is an underpass and $j$ is an overpass, the colors $b,b^{\prime}$ are shifted up by $1$ under $E_{c}$ and we get:

[TABLE]

Finally if $j$ is an underpass instead, colors $b,b^{\prime}$ are shifted down by $1$ and:

[TABLE]

We see that those factors match with the ones in Equation (29) and (30) considering $\varepsilon(l)=+1$. If $l$ corresponds to a negative crossing, only the $\nu(l)$ factor is changed. The computation of the $\nu(l)$ factors is similar and left to the reader.

There is now just one factor to be considered: the factor $F_{c}=\frac{E_{c}w(c)}{w(c)}$ coming from crossing $c$. Assume that $j$ is a positive overpass, then $E_{c}$ shifts the colors $a_{c}^{\prime}$ and $b_{c}$ up by one and leaves colors $a_{c},b_{c}^{\prime}$ invariant. Also here $E_{c}$ shifts $k_{c}$ up by one. We get

[TABLE]

and

[TABLE]

Thus $F_{c}=\mu(c)\nu(c)$ matches with the formula of Proposition 4.5. The other possibilities for $j$ (negative overpass, positive underpass, negative underpass) yield similar computations and are left to the reader. ∎

Recall that the annihilator ideal $\mathrm{Ann}(w_{D})$ is a left ideal of the ring $\mathbbm{Q}[q,Q,Q_{\vec{c}}]\langle E,E_{\vec{c}}\rangle$ where $Q_{\vec{c}}=(Q_{0},\dots,Q_{c(D)})$ and $E_{\vec{c}}=(E_{0},\dots,E_{c(D)})$. Let $\mathrm{Ann}_{\mathrm{rat}}(w_{D})$ denote the corresponding ideal of the ring $\mathbbm{Q}(q,Q,Q_{\vec{c}})\langle E,E_{\vec{c}}\rangle$. Let $R$, $R_{c}$ (for $c=1,\dots,c(D)$) and $R_{0}$ denote the expressions on the right hand side of Equations (28), (27) and (29), (30) respectively.

Proposition 4.7**.**

The ideal $\mathrm{Ann}_{\mathrm{rat}}(w_{D})$ is generated by the set

[TABLE]

Below, we will need to specialize our operators to $q=1$. To make this possible, we introduce the subring $Q_{\mathrm{loc}}(q,Q,Q_{c})$ of the field $\mathbbm{Q}(q,Q,Q_{c})$ that consists of all rational functions that are regular (i.e., well-defined) at $q=1$.

Let $\mathrm{Ann}_{\mathrm{rat},\mathrm{loc}}(w_{D})=\mathrm{Ann}_{\mathrm{rat}}(w_{D})\cap\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{c})\langle E,E_{c}\rangle$ denote the left ideal of the ring $\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{c})\langle E,E_{c}\rangle$.

Proposition 4.8**.**

The ideal $\mathrm{Ann}_{\mathrm{rat},\mathrm{loc}}(w_{D})$ is generated by the set (31).

Proof.

First, let us note that $\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{\vec{c}})\langle E,E_{\vec{c}}\rangle$ is a subring of $\mathbbm{Q}(q,Q,Q_{\vec{c}})\langle E,E_{\vec{c}}\rangle.$

Indeed, if $P(q,Q,Q_{\vec{c}})$ is in $\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{\vec{c}})$ then $EP(q,Q,Q_{\vec{c}})=P(q,q^{-1}Q,Q_{\vec{c}})E$ is also in $\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{\vec{c}})\langle E,E_{\vec{c}}\rangle,$ as the denominator of $P(q,q^{-1}Q,Q_{\vec{c}})$ evaluated at $q=1$ is the same as that of $R(q,Q,Q_{\vec{c}})$. The same can said for multiplication by one of the $E_{\vec{c}}$’s.

Secondly, it is easy to see that the elements $R(q,Q,Q_{\vec{c}})$ and $R_{c}(q,Q,Q_{\vec{c}})$ (for $c=1,\dots,c(D)$) are in $\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{\vec{c}})$. Let $I$ be the left $\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{\vec{c}})\langle E,E_{\vec{c}}\rangle$ ideal generated by those elements.

Let us order monomials in $E$ and the $E_{\vec{c}}$’s using a lexicographic order. We claim that $I$ contains a monic element in each non zero $(E,E_{\vec{c}})$-degree. Indeed, if $E-R(q,Q,Q_{\vec{c}})$ is one of the above described generators and $(\alpha,\beta_{\vec{c}})\in\mathbbm{N}^{c(D)+2}$, multiplying by $E^{\alpha}E_{\vec{c}}^{\beta_{\vec{c}}}$ on the left we get that $I$ contains an element of the form $E^{\alpha+1}E_{\vec{c}}^{\beta_{\vec{c}}}-\tilde{R}(q,Q,Q_{\vec{c}})E^{\alpha}E_{\vec{c}}^{\beta_{\vec{c}}}$ where $\tilde{R}(q,Q,Q_{\vec{c}})\in\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{\vec{c}})$. Using also the generators $E_{\vec{c}}-R_{c}(q,Q,Q_{\vec{c}})$ the claim follows.

Now let $P(q,Q,Q_{\vec{c}},E,E_{\vec{c}})$ be an arbitrary element $\mathrm{Ann}_{\mathrm{rat},\mathrm{loc}}(w_{D})$. We may write

[TABLE]

As $I$ contains a monic element in each non-zero $(E,E_{\vec{c}})$ degree, one may subtrack elements of $I$ to $P(q,Q,Q_{\vec{c}},E,E_{\vec{c}})$ to drop its degree until we get that $P-S\in\mathbbm{Q}_{\mathrm{loc}}(q,Q,Q_{\vec{c}})$ for some element $S\in I$. But $P-S$ must also be in $\mathrm{Ann}_{\mathrm{loc}}(w_{D})$, and as $w_{D}\neq 0$ it must be zero. Thus we can conclude that $I=\mathrm{Ann}_{\mathrm{rat},\mathrm{loc}}(w_{D})$. ∎

5. Matching the annihilator ideal and the gluing equations

5.1. From the annihilator of the state summand to the

gluing equations variety

In the previous sections we studied the gluing equations variety $\mathcal{G}_{D}$ of a knot diagram $D$ and the state summand $w_{D}(n,k)$ of the colored Jones polynomial of $K$. In this section we compare the annihilator ideal of the summand with the defining ideal of the gluing equations variety, once we set $q=1$, and conclude that they exactly match. Let us abbreviate the evaluation of a rational function $f(q)$ at $q=1$ by $\mathrm{ev}_{q}f(q)=f(1)$.

Consider the map $\psi$ defined by:

[TABLE]

where $\mathbbm{C}[\mathcal{G}_{D}]$ denotes the coordinate ring of the affine variety $\mathcal{G}_{D}$ and $w_{\lambda}^{-1/2}$ is the element of $\mathbbm{C}[\mathcal{G}_{D}]$ described in Proposition 2.12.

The main result which connects the quantum invariant with the classical one can be summarized in the following.

Theorem 5.1**.**

(a)* We have:*

[TABLE]

(b)* If $P(q,Q,E)\in\varphi(\mathrm{Ann}(F)\cap\mathbbm{Q}[q,Q]\langle E,E_{k}\rangle)$ as in (7), then $P(q,Q,E)$ annihilates the colored Jones polynomial and $P(1,w_{\mu},w_{\lambda}^{-1/2})=0$.*

Proof.

Recall the generators of the annihilator ideal $\mathrm{Ann}_{\mathrm{rat},\mathrm{loc}}(w_{D})$ given by Equation (31), as well as the functions $L_{k}-1$ for $k=0,\dots,c(D)$ of the coordinate ring of $\mathcal{G}_{D}$ defined in Section 2.7. We will match the two.

For an arc of the diagram with color $a$, let $Q_{a}$ be the multiplication by $q^{a}$. We claim that $\varphi(Q_{a})=z_{a}$, the corresponding arc parameter. Indeed, for the arc $[1,2]$ we have $\varphi(Q_{0})=w_{0}$ is the arc parameter of the arc $[1,2]$, and going from arc $[k-1,k]$ to $[k,k+1]$ we shift the multiplication operator by $Q_{c}^{\pm 1}$ and the arc parameter by $w_{c}^{\pm 1}$, depending on whether $k$ is an over- or underpass.

By Equations (28) and (27):

[TABLE]

If $c$ is a crossing with labels $j<j^{\prime}$, and $j$ is an overpass, we have by Equation (29):

[TABLE]

where

[TABLE]

if $c$ is a positive crossing for example. Similarly by Equation (30), if $j$ is an underpass, then

[TABLE]

Comparing $(\mathrm{ev}_{q}\circ\varphi)(E-R(q,Q,Q_{\vec{c}}))$ with Equation (23), we get that

[TABLE]

Comparing $(\mathrm{ev}_{q}\circ\varphi)(E_{0}-R_{0}(q,Q,Q_{\vec{c}}))$ with Equation (22), we get that

[TABLE]

Finally, if $c$ is a crossing with labels $j<j^{\prime}$, comparing $(\mathrm{ev}_{q}\circ\varphi)(E_{c}-R_{c}(q,Q,Q_{\vec{c}}))$ with Equation (21), we get that

[TABLE]

if $j$ is an overpass, while if $j$ is an underpass we get that

[TABLE]

Thus the image of the generators of the ideal $\mathrm{Ann}_{\mathrm{rat},\mathrm{loc}}(w_{D})$ by $\mathrm{ev}_{q}\circ\varphi$ are generators of the ideal $I_{D}$. This proves part (a) of Theorem 5.1. Part (b) follows from part (a) and Equation (7). ∎

5.2. Proof of Theorem 1.1

Proof.

Fix a labeled, oriented planar projection $D$ of $K$. Then, the certificate recursion $\hat{A}^{c}_{D}(q,Q,E)$ annihilates the colored Jones polynomial, as this is true for all $q$-holonomic sums (5). This concludes part (a).

For part (b), Theorem 5.1 implies that $\hat{A}^{c}_{D}(1,w_{\mu},w_{\lambda}^{-1/2})=0\in\mathbbm{C}[\mathcal{G}_{D}]$. In other words, the function $\hat{A}^{c}_{D}(1,w_{\mu},w_{\lambda}^{-1/2})$ in the coordinate ring of $\mathcal{G}_{D}$ is identically zero. Since this is true for every labeled, oriented diagram $D$ of a knot $K$, this concludes part (b) of Theorem 1.1. ∎

Acknowledgements

S.G. was supported in part by DMS-18-11114. S.G. wishes to thank Dylan Thurston for enlightening conversations in several occasions regarding the octahedral decomposition of knot complements and the state sum formulas for the colored Jones polynomial and Christoph Koutchan for enlightening conversations on $q$-holonomic functions. The paper was conceived and completed while both authors were visiting the Max-Planck Institute for Mathematics in Bonn. The authors wish to thank the institute for its hospitality.