A diagrammatic approach to the AJ Conjecture
The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the polynomial), with a classical invariant, namely the defining polynomial of the character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the -polynomial (after we set , and excluding those of -degree zero) coincides with those of the -polynomial. In this paper, we introduce a version of the -polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the -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 -matrix state sum formula for the colored Jones polynomial, and its certificate.
Key words and phrases: Knot, planar projection, planar diagram, Jones polynomial, colored Jones polynomial, AJ Conjecture, -holonomic sequences, certificate, holonomic modules, gluing equations, character variety.
- 1 Introduction
2 Knot diagrams, their octahedral decomposition and
their gluing equations
- 2.1 Ideal triangulations and their gluing equations
- 2.2 Spines and gluing equations
- 2.3 The octahedral decomposition of a knot diagram
- 2.4 The spine of the -triangulation of a knot diagram and its gluing equations
- 2.5 Labeled knot diagrams
- 2.6 Analysis of triangle and shingle relations
- 2.7 Analysis of big region equations
- 2.8 Formulas for the loop equations
- 2.9 A square root of the holonomy of the longitude
- 3 -holonomic functions, creative telescoping and certificates
- 4 The colored Jones polynomial of a knot
- 5 Matching the annihilator ideal and the gluing equations
1.1. The colored Jones polynomial and the AJ Conjecture
The Jones polynomial of a knot [Jones] is a powerful knot invariant with deep connections with quantum field theory, discovered by Witten [Witten:CS]. The discoveries of Jones and Witten gave rise to Quantum Topology. An even more powerful invariant is the colored Jones polynomial of a knot , a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. Since the dependence of the colored Jones polynomial on the variable plays no role in our paper, we omit it from the notation. The colored Jones polynomial determines the Alexander polynomial [B-NG], is conjectured to determine the volume of a hyperbolic knot [K95, K97, MM], is conjectured to select two out of finitely many slopes of incompressible surfaces of the knot complement [Ga:slope], and is expected to determine the character variety of the knot, viewed from the boundary [Ga:AJ]. The latter is the AJ Conjecture, which is the focus of our paper.
The starting point of the AJ Conjecture [Ga:AJ] is the fact that the colored Jones polynomial of a knot is -holonomic [gale2], that is, it satisfies a nontrivial linear recursion relation
where for all . We can write the above equation in operator form as follows where is an element of the ring where are the operators that act on sequences of functions by:
Observe that the set
is a left ideal of , nonzero when is -holonomic. Although the latter ring is not a principal left ideal domain, its localization is, and cleaning denominators allows one to define a minimal -order, content-free element which annihilates the colored Jones polynomial.
On the other hand, the -polynomial of a knot [CCGLS] is the defining polynomial for the character variety of 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 of positive -degree coincide with those of . The AJ Conjecture is known for most 2-bridge knots, and some 3-strand pretzel knots; see [Le:2bridge] and [LZ:AJ].
Let us briefly now discuss the -holonomicity of the colored Jones polynomial [gale2]: this follows naturally from the fact that the latter can be expressed as a state-sum formula using a labeled, oriented diagram of the knot, placing an -matrix at each crossing and contracting indices as described for instance in Turaev’s book [Tu:book]. For a diagram with crossings, this leads to a formula of the form
where the summand is a -proper hypergeometric function and for fixed , the support of the summand is a finite set. The fundamental theoreom of -holonomic functions of Wilf-Zeilberger [WZ] concludes that is -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 or less) number of crossings and small color (eg, ).
Aside from quantum topology, and key to the results of our paper, is the fact that a planar projection of a knot gives rise to an ideal octahedral decomposition of its complement minus two spheres, and thus to a gluing equations variety and to an -polynomial reviewed in Section 2 below. In [Kim:octI], Kim-Kim-Yoon prove that coincides with the -polynomial of , and in [Kim:octIII] Kim-Park prove that is, up to birational equivalence, invariant under Reidemeister moves, and forms a diagrammatic model for the decorated 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 [Kim:octI] and [Kim:octIII].
1.2. -holonomic sums
To motivate our results, consider a sum of the form
where and and is a proper -hypergeometric function with compact support for fixed . Then is -holonomic but more is true. The annihilator
of the summand is a -holonomic left ideal where and are operators, each acting in one of the variables with the obvious commutation relations (operators acting on different variables commute and the ones acting on the same variable -commute). Consider the map
It is a fact (see Proposition 3.2 below) that
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 [AB, Zeil:creative]. If and denotes generators of the left and the right hand side of (7), it follows that is a right divisor of . We will call the latter the certificate recursion of obtained from (5).
In a sense, the certificate recursion of is more natural than the minimal order recursion and that is the case for holonomic -modules and their push-forward, discussed for instance by Lairez [Lairez].
What is more important for us is that if one allows presentations of of the form (5) where is allowed to change by for instance, consequences of the -binomial identity, then one can obtain an operator which is independent of the chosen presentation.
1.3. Our results
Applying the above discussion to (4) with , allows us to introduce the certificate recursion of the colored Jones polynomial, which depends on a labeled, oriented planar diagram of a knot. We can also define to be the left gcd of the elements in the local ring , lifted back to .
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 -matrix state sum formula for the colored Jones polynomial, and its certificate.
For every knot ,
(a) divides .
(b) Every irreducible factor of of positive -degree is a factor of .
The -polynomial has only been computed in a handful of cases, see [GS:twist], [GM:pretzel], [GK:pretzel] and [GK:74]. In all cases where is known, it is actually obtained from certificates and in that case .
Is it true that for any knot , one has ?
Is it true that the certificate recursion of a planar projection of a knot is invariant under Reidemester moves on ?
A positive answer to the latter question is a quantum analogue of the fact that the gluing equation variety associated to a diagram is independent of , a result that was announced by Kim and Park [Kim:octIII]. 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 -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 of an oriented knot . 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 -matrix per crossing and contracting indices. The summand of this state-sum is -proper hypergeometric and its annihilator defines an ideal in a quantum Weyl algebra, discussed in Section 4. The annihilator ideal is matched when with the gluing equations ideal in the key Proposition 5.1. This matching, implicit in the Grenoble notes of D. Thurston [Thu:grenoble], combined with a certificate (which is a quantum version of the projection map from gluing equations variety to ), and with the fact that the gluing equation variety sees all components of the character variety [Kim:octI], 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 [And, Dimofte]. 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 [And].
Finally, our proof of Theorem 1.1 does not imply any relation between the Newton polygon of the polynomial and that of . If the two Newton polygons coincided, the Slope Conjecture of [Ga:slope] would follow, as was explained in [Ga:quadratic]. 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 of a 3-manifold with cusps, Thurston’s gluing equations (one for each edge of ) give a way to describe the hyperbolic structure on and its deformation if is hyperbolic [Th, NZ]. The gluing equations define an affine variety , 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)
satisfies the equations
and every solution of (8) uniquely defines a triple of shapes of a tetrahedron. Note that the shapes of the tetrahedron , , or lie in , and that they are uniquely determined by . When we talk about assigning a shape to a tetrahedron below, it determines shapes and as in Figure 1.
Given an ideal triangulation with tetrahedra, assign shapes for to each tetrahedron. If is an edge of the corresponding gluing equation is given by
where is the set of all tetrahedra that meet along the edge , and is the shape parameter corresponding to the edge of . The gluing equation variety is the affine variety in the variables defined by the edge gluing equations, for all edges of . Equivalently, it is the affine variety in the variables 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) representations of the fundamental group of the underlying 3-manifold . The construction of decorated representations from solutions to the gluing equations appears for instance in Zickert’s thesis [Zickert:thesis] and also in [GGZ]. 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 from the universal cover to the 3-dimensional hyperbolic space . Since the orientation preserving isometries of are in , this in turn gives rise to a representation of the fundamental group , 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 denote the augmented character variety of . Thus, we get a map:
So far, can have boundary components of arbitrary genus. When the boundary consists of a single torus boundary component, and is a simple closed curve on , the holonomy of an augmented representation gives a regular function . Note that for a decorated representation , the set of squares of the eigenvalues of is given by . Once we fix a pair of meridian and longitude of the boundary torus, then we get a map
The defining polynomial of the 1-dimensional components of the above map is the -polynomial of the 3-manifold . Technically, this is the -version of the -polynomial and its precise relation with the -version of the -polynomial (as defined by [CCGLS]) is discussed in detail in Champanerkar’s thesis [Ch]; 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 , and hence the gluing equations of the triangulation may not detect some factors of the -polynomial. In fact, when the boundary of consists of tori, the image of (9) always misses the components of abelian 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 knot with an edge of valency one, and for that triangulation, 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 in a component of that is normal with respect to this triangulation, it intersects the triangles of 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 or depending whether the segment goes left or right. The holonomy of is then:
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 -skeleton. Because of this reason, we discuss the gluing equations of an ideal triangulation in terms of its spine. In that case, edges of are dual to 2-cells of the spine, and give rise to gluing equations. Recall that a spine of is a CW-complex embedded in , such that each point of has a neighborhood homeomorphic to either , where is the -shaped graph or to the cone over the edges of a tetrahedron, and such that is homeomorphic to . 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 , 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 , and the other bear the parameters according to the cyclic ordering (see Figure 1). Edge equations translate into region equations, the region equation associated to the region being:
For a path on the spine that is in normal position with respects to , it intersects each region in a collection of segments . The holonomy of the segment is
where left and right corners are defined as in Figure 2, and the holonomy of is
2.3. The octahedral decomposition of a knot diagram
In this section we fix a diagram in of an oriented knot . By diagram, we mean an embedded 4-valent graph in the plane, with an overcrossing/undercrossing choice at each vertex. Let and denote the set and the number of crossings of . In this section as well as the remainder of the paper, an arc of will be the segment of the diagram joining two successive crossings of . 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 and the set of underpasses by . An overarc (resp. underarc) will be the portion of the knot joining two successive underpasses (resp. overpasses). An overarc of may pass through some number of crossings of , doing so as the upper strand each time.
Given a diagram of the knot with crossings, let be some ball lying above the projection plane and 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 [snappy, Weeks], yields a decomposition of into ideal octahedra. The decomposition works as follows: at each crossing of , 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 and the two other middle vertices down towards . One can then patch all these octahedra together to get a decomposition of . We refer to [Kim:octI] as well as [Thu:grenoble] for figures and more details on this construction.
From the octahedral decomposition of , one can get an ideal triangulation of simply by splitting the octahedra further into tetrahedra. There are two natural possibilities for this splitting, as one can cut each octahedra into either or tetrahedra as shown in Figure 3. We will be interested in the decomposition where we split each octahedra into tetrahedra, obtaining thus a decomposition of into tetrahedra. We denote this ideal triangulation by , and we call it the “-triangulation of ”.
Since the inclusion map is an isomorphism on fundamental groups, a solution to the gluing equations of gives rise to a decorated representation of the knot complement.
2.4. The spine of the -triangulation of a knot diagram and its gluing equations
Let denote the gluing equation variety of . To write down the equations of , 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 [Kim:octI].
Figure 4 shows a picture of the spine near a crossing of . The spine contains vertices near each crossing of and can be described as follows:
First we embed in 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 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 to the rest of the spine. The regions of the spine are then of types:
An upper/lower triangle region for each crossing, and in total.
For each region of one gets an horizontal region in the spine; we call these big regions, in total.
The boundary of a neighborhood of 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, in total. Similarly, the lower shingle regions are in correspondance with underarcs, and there is also of them.
We now assign shape parameters to each vertex of the spine as shown in Figure 4. There are shape parameters for each crossing : a central one which we call and others: standing for lower-in, lower-out, upper-in and upper-out. When the crossing we consider is clear, we will sometimes write dropping the index .
Note that the assignment of shape parameters is such that the main version of the parameter lies on a corner of a triangle region, while the auxiliary are prescribed by the cyclic ordering induced by the boundary of .
We can now write down the gluing equations coming from the -spine:
The upper/lower triangle equations are (in the notation of Figure 4)
The upper/lower shingle equations. Consider an upper shingle region corresponding to an overarc going from some crossing labelled to the crossing , going through crossings as overpasses. Then the shingle region has one corner for each of its ends, and corners for each overpasses, as explained in Figure 5. We get:
The upper/lower shingle equations have the equivalent forms, respectively:
Grouping together shape parameters coming from the same vertex and using , we get:
and then, using Equation (11):
Similarly for a lower shingle region corresponding to an underarc running from crossing to crossing , one gets an equation:
which simplifies to (13). ∎
Figure 6 shows a top-view of the -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 gives rise to corners in the corresponding big region. For each region of , we get a big region equation of the form
where the corner factors are prescribed by the rule shown in Figure 6.
Below, we will denote the triangle, region and shingle equations by , and respectively. The above discussion defines the gluing equations variety as an affine subvariety of defined by
We now express the holonomies and of the meridian and zero winding number longitude in terms of the above shape parameters. Note that if is not the unknot, it is always possible to find in the diagram of an underpass that is followed by an overpass that corresponds to a different crossing of . We then name those two crossings and . Assume that the meridian is positioned as shown in Figure 7. Then the rule described in Section 2.1 gives us the following holonomy:
As , we get:
Finally, we turn to the holonomy of a longitude. We first compute the holonomy of the longitude corresponding to the blackboard framing of the knot. We can represent this longitude on the diagram as a right parallel of . 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
and each lower shingle segment using the convention
We can actually ignore the signs as there are segments, an even number.
As Figure 8 shows, we get:
The last product is over the set of crossings of , and for simplicity we do not indicate the dependence of the variables on the crossing . Let be the longitude with zero winding number with . The winding number of the blackboard framing longitude is the writhe of the diagram , which can be computed by , where and are the number of positive and negative crossings of the diagram. We then have and thus
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 is a planar diagram of an oriented knot and that we have chosen two special crossings and that are successive in the diagram, such that such crossing corresponds to an underpass and crossing to an overpass. This choice determines a labeling of crossings of as follows.
Following the knot, we label the other crossings Note that as the knot passes through each crossing twice, each crossing of gets two labels . 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 and (or and ). We write for the arc joining crossings and .
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 . We have the following:
In , each of the variables are monomials in the variables and .
Fix a labeled knot diagram as in Section 2.5. Before eliminating variables, we start by assigning to each arc of the diagram a new parameter . These parameters are expressed in terms of the previous parameters by the following rules:
We recall that in the above (resp. ) is the set of overpasses (resp. underpasses) in the diagram . Also, given integers with , we denote
Note that the arc parameters are all clearly monomials in and the ’s.
We claim that each of the shape parameters are monomials in the ’s and . This will imply the proposition. Indeed, let be an arc of . Then we claim that:
Note by definition. If is an underpass, the formula
matches with the upper shingle equation expressing in terms of . Indeed, if is the underpass coming immediately after underpass , Equation (12) says:
As crossings correspond to overpasses and to an underpass, we also have
By induction, we find that for any underpass .
The second case is then a consequence of the lower triangle equation , and the fact that as is an underpass.
Note that by Equation (16), so the fourth case is valid for the arc . 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 , and . ∎
In the rest of the paper, we will often use the arc parameters defined above to express equations in
For instance, thanks to Proposition 2.2, we can rewrite the big region equations as equations , where is expressed in terms of the variables only.
Although the arc parameters are just monomials in the variables, they are helpful for writing down the equations defining in a more compact way. When the choice of a crossing is implicit, we introduce a simplified notation for the parameters associated to arcs neighboring . We will write 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:
For instance, we get a new expression of the holonomy of the longitude:
With the convention of Remark 2.3, the holonomy of the zero-winding number longitude is expressed by:
By Equation (17) we have: