Nahm sums, stability and the colored Jones polynomial
 Stavros Garoufalidis^{1}Email author and
 Thang TQ Lê^{1}
https://doi.org/10.1186/2197984721
© Garoufalidis and Lê; licensee Springer. 2015
Received: 18 March 2014
Accepted: 18 March 2014
Published: 20 January 2015
Abstract
Nahm sums are qseries of a special hypergeometric type that appear in character formulas in the conformal field theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in the quantum knot theory  we prove the stability of the coefficients of the colored Jones polynomial of an alternating link and present a Nahm sum formula for the resulting power series, defined in terms of a reduced diagram of the alternating link. The Nahm sum formula comes with a computer implementation, illustrated in numerous examples of proven or conjectural identities among qseries.
MSC
Primary 57N10; Secondary 57M25.
Keywords
1 Background
The colored Jones polynomial of a link is a sequence of Laurent polynomials in one variable with integer coefficients. We prove in full a conjecture concerning the stability of the colored Jones polynomial for all alternating links.
A weaker form of stability (zero stability, defined below) for the colored Jones polynomial of an alternating knot was conjectured by Dasbach and Lin. The zero stability is also proven independently by Armond for all adequate links [1], which include alternating links and closures of positive braids, see also [2]. The advantage of our approach is that it proves stability to all orders and gives explicit formulas (in the form of generalized Nahm sums) for the limiting series, which in particular implies convergence in the open unit disk in the qplane and allow for the study of their redial asymptotics.
Stability was observed in some examples by Zagier, and conjectured by the first author to hold for all knots, assuming that we restrict the sequence of colored Jones polynomials to suitable arithmetic progressions, dictated by the quasipolynomial nature of its qdegree [3, 4]. Zagier asked about modular and asymptotic properties of the limiting qseries. In a similar direction, Habiro asked about zero stability of the cyclotomic function of alternating links in [5].
Our generalized Nahm sum formula comes with a computer implementation (using as input a planar diagram of a link), and allows the study of its asymptotics when q approaches radially a root of unity. Our Nahm sum formula is reminiscent to the cohomological Hall algebra of motivic DonaldsonThomas invariants of KontsevichSoibelman [6] and complement recent work of Witten [7] and DimofteGaiottoGukov [8].
1.1 Nahm sums
We will abbreviate (x;q)_{ n } by (x)_{ n }.
where A is a positive definite even integral symmetric matrix and .
Nahm sums appear in character formulas in the conformal field theory, and define analytic functions in the complex unit disk q<1 with interesting asymptotics at complex roots of unity, and with sometimes modular behavior. Examples of Nahm sums are the seven famous, mysterious qseries of Ramanujan that are nearly modular (in modern terms, mock modular). For a detailed discussion, see [11]. Nahm sums give rise to elements of the Bloch group, which governs the leading radial asymptotics of f(q) as q approaches a complex root of unity. Nahm’s conjecture concerns the modularity of a Nahm sum f(q), and was studied extensively by Zagier, VlasenkoZwegers and others [12, 13].
of power series in q with integer coefficients and bounded below minimum degree. In the remaining of the paper, by Nahm sum we will mean a generalized Nahm sum. The paper is concerned with a new source of Nahm sums that originate in the quantum knot theory.
1.2 Stability of a sequence of polynomials
For let mindeg_{ q } f(q) denote the smallest j such that a _{ j }≠0 and let coeff(f(q),q ^{ j })=a _{ j } denote the coefficient of q ^{ j } in f(q).
Definition 1
if
(and in particular, coeff(f _{ n }(q),q ^{ j })=coeff(f(q),q ^{ j })) for all n>N _{ j }.
Remark 1
Although for every integer j we have , it is not true that .
Definition 2
It is easy to see that the pointwise sum and product of kstable sequences are kstable.
1.3 Stability of the colored Jones function for alternating links
When K is an alternating link, the lowest degree of J _{ K,n }(q) is known and the lowest coefficient is ±1 (see [16, 17] and Section 7). We divide J _{ K,n }(q) by its lowest monomial to obtain . Although , we have ; see [18].
Our main results link the colored Jones polynomial and its stability with Nahm sums. The first part of the result, with proof given in Section 9, is the following:
Theorem 1
For every alternating link K, the sequence is stable and its associated klimit Φ _{ K,k }(q) and series F _{ K }(x,q) can be effectively computed from any reduced, alternating diagram D of K.
Let us give some remarks regarding Theorem 1.
Remark 2
If one uses the new normalization where with J _{Unknot,n }(q)=1, the above theorem still holds. The new F _{ K }(x,q) is equal to the old one times (1−q)/(1−x).
Remark 3
If is the mirror image of K, then . If K is alternating, then so is . Hence, applying Theorem 1 to , we see that similar stability result holds for the head of the colored Jones polynomial of alternating link.
Remark 4
The weaker zero stability (conjectured by Dasbach and Lin) is proven independently by Armond [2]. In [2], zero stability is proved for all Aadequate links, which include all alternating links, but no stability in full is proven there, nor any formula for the zero limit is given. As we will see, the proof of stability in full is more complicated than that of zero stability and occupies the more difficult part of our paper, given in Sections 8 to 10.
Remark 5
A sharp estimate regarding the rate of convergence of the stable sequence is given in Theorem 4.
1.4 Explicit Nahm sum formulas for the zero limit and one limit
Throughout this subsection, D is a reduced diagram of a nonsplit alternating link K with c crossings.
1.4.1 Laplacian of a graph
In this paper, a graph is a finite onedimensional CWcomplex. A plane graph is a graph Γ (with loops and multiple edges allowed) together with an embedding of Γ into . A plane graph Γ gives rise to a polygonal complex structure of S ^{2}, and its set of vertices, set of edges, and set of polygons are denoted respectively by , and .
The adjacency matrix Adj(Γ) is the matrix defined such that Adj(Γ)(v,v ^{′}) is the number of edges connecting v and v ^{′}. Let Deg(G) be the diagonal matrix such that Deg(Γ)(v,v) is the degree of the vertex v, i.e. the number of edges incident to v, with the convention that each loop edge at v is counted twice.
1.4.2 Graphs associated to a reduced alternating nonsplit link diagram D
This is the usual checkerboard coloring of the regions of an alternating link diagram, used already by Tait. When we rotate the overcrossing arc at a crossing counterclockwise (resp. clockwise), we swap an Atype (resp. Btype) angle. Note that orientation dose not take part in the definition of Aangles and Bangles.
Let D ^{∗} be the dual of the plane graph D. Since D has a checkerboard coloring of its faces, it follows that D ^{∗} has a coloring of its vertices by A or B. Thus, give a bipartite structure on where and are the sets of Acolored and Bcolored vertices of .
Since the degree of each vertex of D is 4, each polygon of D ^{∗} is a quadrilateral, having four vertices, two of which are A vertices and two are B vertices. Moreover, the vertices of each quadrilateral alternate in color. Connect the two B vertices of each quadrilateral of D ^{∗} by a diagonal inside that quadrilateral, and call it a edge. The Tait graph of D is defined to be the plane graph whose set of vertices is , and whose set of edges is the set of egdes. The plane graph Tait graph totally determines the alternating link K up to orientation. The graph can be defined for any link diagram, and is studied extensively, see e.g. [20][22].
Note that for a vertex , its degrees in D ^{∗} and in are the same.
1.4.3 The lattice and the cone
Fix an Avertex of D ^{∗} and call it v _{ ∞ }. We will focus on , the lattice of rank c+2 freely spanned by the vertices of D ^{∗}. Let , a sublattice of Λ of rank c+1.
An element x∈Λ _{0} is admissible if e(x)≥0 for every edge . The set Adm⊂Λ _{0} of all admissible elements is the intersection of Λ _{0} with a rational convex cone in .
Note that a priori is a matrix, and is considered as a matrix in the righthand side of (4) by the trivial extension, i.e. in the extension, any entry outside the block is 0.
Remark 6
Although Q(λ) and L(λ) take value in , we later show that . While Q,L depend only on D, the set Adm depends on the choice of an Avertex v _{ ∞ }.
Examples that illustrate the above definitions are given in Section 1.6.
1.4.4 Nahm sum for the zero limit
The next theorem is proven in Section 7.
Theorem 2.
The generalized Nahm sum on the righthand side is regular and belongs to .
A categorification of the above theorem was given recently by Rozansky [23]. Here are two consequences of this explicit formula. The next corollary is proven in Section 7.4.
Corollary 1
For every alternating link K, is analytic in the unit disk q<1.
The next corollary is shown in Section 13.
Corollary 2
If the reduced Tait graphs of two alternating links K _{1},K _{2} are isomorphic as abstract graphs, then they have the same zero limit, .
Here the reduced Tait graph is obtained from by replacing every set of parallel edges by an edge; and two edges are parallel if they connect the same two vertices. This corollary had been proven by Armond and Dasbach: in [24], it is proved that if two alternating links have the same reduced Tait graph, and the zero limit of the first link exists, then the zero limit of the second one exists and is equal to that of the first one. In Section 13, we will derive Corollary 2 from the explicit formula of Theorem 2.
We end this section with a remark on normalizations.
Remark 7
where ⊔ and ♯ denotes the disjoint union and the connected sum respectively.
1.4.5 The one limit
The next theorem is proven in Section 12.2.
Theorem 3.
where Admv is the set of all admissible x such that p(x)=0 for every incident to v.
Remark 8
The fact that the series (6) is convergent is not obvious. It follows from the fact that we can separate the sum over admissible states λ to those which are not onebounded and those which are onebounded. Here, a state λ is oneunbounded if ; see Definition 5. It is easy to see that the contribution of the oneunbounded states in (6) forms a convergent series. For the onebounded states s, one uses a decomposition theorem s=m s _{ P }+s ^{′} discussed in Example 2. Then, the contribution of such a state to (6) comes with minimum degree Q(s ^{′})+L(s ^{′})+m. This implies that the contribution of the onebounded states in (6) forms a convergent series too.
For an example illustrating Theorems 2 and 3, see Section1.6.
1.5 qholonomicity
for all where for all j and c _{ d }≠0.
The next theorem (proven in Section 11) shows the qholonomicity of Φ _{ K,n }(q) for an alternating link, and gives a sharp improvement of the rate of convergence in the definition of stability.
Theorem 4.
 (a)
for all k when n is sufficiently large (depending on k).
The lowest q exponent in Equation 9 is a quadratic function of k. This result is sharp when K=4_{1} knot [28, 29].
Question 1
Does F _{ K,n }(x,q) uniquely determine the sequence for the case of knots?
1.6 Applications: qseries identities
To compute , proceed as follows:

Checkerboard color the regions of D with A or B with the unbounded region colored by A.

Assign variables a,b,c to the three Bregions and e,f to the two bounded Aregions, and assign 0 to the unbounded Aregion. Let λ=(a,b,c,d,e)^{ T }.

Construct a square matrix (and a corresponding quadratic form Q(λ)) which consists of four blocks: BBblock, ABblock, BAblock and AAblock. On the BB, AB and BAblocks, we place the adjacency matrix of the corresponding regions: the adjacency number between two distinct Bregions is the number of common vertices, whereas the adjacency number between an Aregion and a Bregion is the number of common edges. In the case when two regions share common vertices, the adjacency number is the number of common vertices. On the AAblock, we place the diagonal matrix whose diagonal entries are the number of sides of each Aregion.

We construct a linear form L(λ) in λ where the coefficient of each Bvariable a,b,c is one, and the coefficient of each Avariable d,e is half the number of the sides of the corresponding region minus 1.Explicitly, with the conventions of Figure 2, we have
The above identity has been proven by ArmondDasbach. A detailed list of identities for knots with knots with at most 8 crossings is given in Appendix 14.
1.7 Extensions of stability
The methods that prove Theorem 1 are general and apply to several other circumstances of qholonomic sequences that appear in quantum topology. We will list two results here.
Theorem 5.
If K is a positive link, then is stable and the corresponding limit F _{ K }(x,q) is obtained by a Nahm sum associated to a positive downwards diagram of K. Moreover, for every we have .
Some results related to the zero stability of a class of positive knots are obtained in [32].
Theorem 6.
For every admissible γ, the sequence is stable, and its limit is given by a Nahm sum.
The proof of the above theorem follows easily from the fact that the quantum 6jsymbol is given by a onedimensional sum of a qproper hypergeometric summand, and the sum is already centered. The analytic and arithmetic properties of the corresponding Nahm sum will be discussed in forthcoming work [28, 29].
1.8 Plan of the proof
The strategy to prove Theorems 2 and 1 is the following:
We begin with the Rmatrix state sum for the colored Jones polynomial, reviewed in Sections 2.2 to 2.4.
We center the downward diagram, its corresponding states and their weights in Section 4.1.
We factorize the weights of the centered states as the product of a monomial and an element of in Section 4.1. The advantage of using centered states is that the lowest qdegree of their weights is the sum of a quadratic function Q(s) of s with a quadratic function of n.
Although Q(s) is not a positive definite quadratic form, in Section 5.3 we show that Q(s) is copositive on the cone of the centered states. The proof uses the combinatorics of alternating downward diagrams, and their centered states, reminiscent to the Kauffman bracket.
In Section 7, we prove the zero stability Theorem 2.
If Q(s) were positive definite, then it would be easy to deduce Theorem 1. Unfortunately, Q(s) is never positive definite, and it always has directions of linear growth in the cone of centered states. In Section 8, we state a partition of the set of kbounded states, and prove stability away from the region of linear growth. Section 9 deals with stability in the region of linear growth.
Section 10 is rather technical, and gives a proof of the key Proposition 4.
Section 11 deduces the qholonomicity of the sequence Φ _{ K,k }(q) of an alternating link from the qholonomicity of the corresponding colored Jones polynomial. As a result, we obtain sharp quadratic lower bounds for the minimum degree of Φ _{ K,k }(q) and sharp bounds for the convergence of the colored Jones polynomial stated in Theorem 4.
In Section 12, we give an algorithm for computing Φ _{ K,k }(q) from a reduced alternating planar projection.
In Section 13, we prove that Φ _{ K,0}(q) is determined by the reduced Tait graph of an alternating link K.
In Section 14, we give some illustrations of Theorems 2 and 1.
1.9 Followup work
The topic of stability (affectionately called head/tail by DasbachLin [36]) has recently attracted a lot of attention. After the papearance of our paper on the arxiv in the late 2011, a number of papers have since been posted. Among them, Hajij gives a skeintheory proof of zero stability for alternating links and some quantum spin networks [37, 38]. Motivated by the qseries of Nahm type, Andrews proves some RogersRamanujan type identities [39]. Vuong and the first author give efficient algorithms to compute the zero limit of an alternating knot [40]. VuongNorin and the first author identify the coefficients of q ^{ k } of the zero limit for k=0,…,3 in terms of graph countings of induced plane subgraphs of the reduced Tait graph of an alternating link [41]. Norin and the first author prove that each coefficient of q ^{ k } in the zero limit is a polynomial of induced plane subgraphs of the reduced Tait graph of an alternating link [42]. Finally, in another direction, qseries of Nahm type were studied by BeemDimoftePasquetti and Kashaev and the first author; see [43, 44].
2 The Rmatrix statesum of the colored Jones polynomial
2.1 Downward link diagram
where D _{1} and D _{2} are diagrams with at least one crossing.
2.2 Link diagrams and states
where as usual the sign of the crossing on the lefthand side of (11) is positive and the sign of the one on the righthand side is negative. For a positive integer n, a state is called nadmissible if the values of r are integers in [ 0,n] and r(v)≥0 for every crossing v. Let S _{ D,n } be the set of all nadmissible states.
Remark 9
Later we will prove that . By definition, S _{ D,n } in 11 correspondence with the set of lattice points of n P _{ D } for a lattice polytope P _{ D } in where c _{ D } is the number of crossings of D.
2.3 Winding number and its local weight
Suppose α is an oriented simple closed curve in the standard plane. By the winding number W(α), we mean the winding number of α with respect to a point in the region bounded by α. Observe that W(α)=1 if α is counterclockwise, −1 if otherwise.
The next lemma is elementary.
Lemma 1
where the sum is over all the local parts of α.
2.4 Local weights, the colored Jones polynomial, and their factorization
Fix a natural number n≥1 and a downward link diagram D.
Note that J _{ K,0}(q)=1 for all links and J _{ K,1}(q ^{−1})/J _{Unknot,1}(q ^{−1}) is the Jones polynomial of K[45]. Since we could not find a reference for the state sum formula (14) in the literature, we will give a proof in the Appendix.
3 Alternating link diagrams and centered states
In this section, we will discuss the combinatorics of alternating diagrams.
3.1 Alternating link diagrams and Ainfinite type
Recall that a link diagram D gives rise to a polygonal complex structure of , and if D is alternating and connected, then the checkerboard coloring with colors A and B at each crossing looks like Figure 1.
If K is nonsplit, then D is a connected graph. If K is split, then D has several connected components. We will say that an alternating diagram D is Ainfinite if the point ∞∈S ^{2} is contained in an Apolygon of every connected subgraph of D. It is clear that by moving the connected components of D around in S ^{2}, we can assume that D is Ainfinite. This will make the colors of different connected components compatible.
We will use the following obvious property of an Ainfinite alternating link diagram: all the Bpolygons are finite, i.e. in .
For example, the lefthanded trefoil given by the standard closure of the braid is Ainfinite, whereas the righthanded trefoil given by the standard closure of the braid is not. Here s _{1} is the standard generator of the braid group in two strands.
3.2 The digraph of an alternating diagram D
Let D be an oriented link diagram. Recall that we consider D also as a graph whose edges are oriented. We say that an edge of D is of type O if it begins as an overpass, and of type U if it begins as an underpass. If D is alternating and one travels along the the link, the edges alternate from type U to type O and viceversa.
i.e., if the crossing is a positive one, then the two left edges incident to it get orientation reversed, and if the crossing is negative, then the two right edges incident to it get orientation reversed.
We will retain the markings A and B for angles and regions of complements of .
Note that after either type of smoothening, the orientation of the edges of is still welldefined.
Remark 10
Doing an Aresolution (resp. Bresolution) on a vertex of is the same as doing an Aresolution (resp. Bresolution) on the original diagram D in the sense of Kauffman [47]. The advantage here, with directed graph for the case of alternating links, is that the resulting graph of any resolution is still oriented.
Part (b) of the following lemma is where Ainfinity is used in an essential way.
Lemma 2
Suppose D is an alternating link diagram, (a) To the right of every oriented edge of is an Apolygon, and to the left of every oriented edge of is a Bpolygon. (b) Suppose D is Ainfinite, then every circle obtained from by after doing Aresolution at every vertex of bounds a polygonal region of type B. Moreover, every such circle is winding counterclockwise, i.e., it has winding number 1. (c) If D is reduced, then each circle in (b) does not selftouch, i.e., the two arcs resulting from the Aresolution at one vertex do not belong to the same circle.
Proof
(a) This follows easily by inspecting the directions of the edges and the markings of the regions at the two types of vertices of . (b) The boundaries of the Bpolygons are exactly the circles obtained from D after doing Aresolution at every vertex of D. Since the infinity region is not a Btype region, every circle does bound a Btype region in the plane . From part (a), it follows that each circle, which is the boundary of a polygonal region of type B, is counterclockwise. (c) This is a wellknown fact. A link diagram having the property that no circle obtained after doing Aresolution at every crossing has a selftouching point is known as an Aadequate diagram. In [17, Prop.5.3], it was proved that every reduced alternating link diagram is Aadequate.
3.3 Centered states
Fix an alternating downward diagram D with c _{ D } crossings and its directed graph . Recall that and denote respectively the set of oriented edges of and the set of vertices of .
It is easy to see that the map (19) is a vector space isomorphism. If r∈S _{ D,n }, i.e., r is nadmissible, then is called nadmissible. Let be the set of all nadmissible centered states.
The following is a reformulation of nadmissibility in terms of centered states.
Lemma 3
Proof.
This follows immediately from the definition, since for any state r and for every vertex v of we have .
It follows that if a centered state is nadmissible, then it is (n+1)admissible.
4 Local weights in terms of centered states
In this section, we will give an explicit formula for the weight of a centered state. It turns out that the state sum of the colored Jones polynomial in terms of centered states has the important property of separation of variables needed in the proof of the stability. See Remark 12.
4.1 Local weights of centered states and their factorization
Note that w _{≻}(X,s) is independent of the sign of the local crossing and takes the same value 1 at all local extrema. Hence, we use the notation w _{≻}(v,s) for the righthand side of (21), where is the involved vertex. The following is a convenient way to rewrite the value of w _{≻}(v,s).
Lemma 4
Proof.
Remark 11
4.2 The functionals P _{0},P _{1},Q,L _{0},L _{1}
Here , where the product is over all local parts of D. Note that . The functionals L _{0},L _{1} are linear forms on and do not depend on n in the sense that the value of each of L _{0},L _{1} will be the same if we consider s as an (n+1)admissible centered state instead of an nstate. The functional Q _{2}=Q+L _{1} is a quadratic form on not depending on n. The two functionals P _{0},P _{1} depend only on n, i.e., if s,s ^{′} are nadmissible centered states, then P _{ i }(s)=P _{ i }(s ^{′}). Hence, we will also write P _{ i }(n) instead of P _{ i }(s), for i=0,1.
Lemma 5
Proof.
which is equal to the right hand side of (25) by identity (27) and the definition of F(x,q,s).
Remark 12
where the sum is over all Aangles α.
5 Positivity of Q _{2}and the lowest degree of the colored Jones polynomial
In this section, we prove the copositivity of Q _{2}:=Q+L _{1} on the cone and derive a formula for the lowest degree of the colored Jones polynomial. As before, we fix a reduced, alternating Ainfinite downward diagram D with c _{ D } crossings.
5.1 A Hilbert basis for : elementary centered states
From its very definition, the set of valued centered states of can be identified with the set of lattice points of a lattice cone in . In general, the set of lattice points of a rational cone is a monoid, and a generating set is called a Hilbert basis which plays an important role in integer programming; see for instance [48, Sec.13] and also [49, Sec.16.4]. Note that every element of a finitely generated additive monoid is an linear combination of a Hilbert basis. Although the natural number coefficients are not unique, this is not a problem for applications.
The goal of this section is to describe a useful Hilbert basis for .
Recall that is a directed graph. Suppose γ is a directed cycle of , i.e., closed path consisting of a sequence of distinct edges e _{1},…,e _{ n } of D such that the ending point of e _{ j } is the starting points of e _{ j+1} (index is taken modulo n) and there is no repeated vertex along the path except for the obvious case where the first vertex is also the last vertex. An example of a cycle of is the boundary of a polygon in the complement of .
Definition 3
For a directed cycle γ of , let s _{ γ } be the function on the set of edges of which assigns 1 to every edge of γ and 0 to every other edge. Such a centered state is called elementary, and γ is called its support. Let denote the (finite set) of all elementary centered states of .
For a polygon , the boundary ∂ p is a directed cycle of , and we will use the notation s _{ p }:=s _{ ∂ p }.
From Lemma 3, we see that s _{ γ } is an nadmissible centered state for every n≥1.
Lemma 6
Proof.
Let s be an valued centered state of . Suppose e is an oriented edge such that s(e)>0. At the ending vertex v of e let e ^{′} and e ^{″} be the two edges which are perpendicular to e. Inspection of Equation 15 shows that v is the starting vertex for both e ^{′} and e ^{″}. Equation 16 shows that s(e ^{′})+s(e ^{″})≥s(e). Hence one of them, say s(e ^{′})>0. This means if e is an edge with s(e)>0, we can continue e to another edge e ^{′} for which s(e ^{′})>0. Repeating this process, we can construct a cycle γ of such that the value of s is positive on any edge of γ. This means s−s _{ γ } is an valued centered state. Induction completes the proof of the lemma.
Remark 13
It is easy to see that any is not a linear combination of the other elements in . Thus there is no redundant element in . Of course is linearly dependent over (or over ), and we will extract a basis from the set later.
5.2 Values of L _{1}and Qon elementary centered states
Suppose γ is a directed cycle of and v is a vertex of γ. Among the four edges of incident to v, the two edges of γ are not two opposite edges because of the orientation constraint, see (15). In other words, at each vertex v, γ is an angle. We say that a vertex v of γ is of type A or B according as the two edges of γ at v form an angle of type A or B. Let N _{ γ,A } be the number of vertices of γ of type A. The fact that D is reduced is used in the proof of part (b) of the next lemma.
Lemma 7
Proof.
 (a)
For a local part X of , let γ _{ X }=γ∩X. Clearly L _{1}(X,s)=0 if γ _{ X }=∅. If X is a small neighborhood of a vertex of , then γ _{ X } is two sides of an angle of γ, and we will smoothen γ _{ X } at the corner to get an oriented smoothed arc. See row 1 and row 2 of Table 4 for various X and smoothened γ _{ X }. In the table, X is a small neighborhood of a vextex. The two edges incident to the vertex with label 1 belong to γ. The marking A or B at one of the angles of X indicates the type of the vertex, which appear in row 3. In row 3, we also indicate the sign of the crossing of X (as it appeared originally in D); this makes the computation of L _{1} easier.
 (b)
Case 1: γ is counterclockwise. From (29), we have L _{1}(s)≥W(γ)=1>0. In this case L _{1} is strictly positive.
Case 2: γ is clockwise. Then L _{1}(s)=−1+N _{ γ,A }/2. We will show that N _{ γ,A }≥2.
If N _{ γ,A }=0, then γ is one of the circles obtained from by doing Aresolution at every vertex. By part (b) of Lemma 2, γ is counterclockwise. Thus, N _{ γ,A }≠0 if γ is clockwise.
Suppose N _{ γ,A }=1, i.e. γ has exactly one vertex of type A, say v; all other vertices of γ are of type B. If one does Aresolution at every vertex of , then γ∖{v} is part of one of the resulting circles, and this circle has a selftouching point at v. This is impossible if the diagram D is reduced, see part (c) of Lemma 2. Thus, N _{ γ,A }≠1.
We have shown that if γ is clockwise then N _{ γ,A }≥2. Hence, L _{1}(s)=−1+N _{ γ,A }/2≥0, and equality happens if and only N _{ γ,A }=2.
Remark 14
We see that for the proof of part (b), we need only the fact that D is Aadequate.
Lemma 8
 (a)
 (b)
It follows that Q(s)≥0, with equality if and only if γ is the boundary of a polygonal region of type B.
Proof.
 (a)
Since Q is defined by an expression with positive coefficients, we have Q(s+s ^{′})≥Q(s)+Q(s ^{′}). (b) Row 6 of Table 4 shows that every vertex of type A of γ contributes 1/2 to the value of Q, while others contribute 0. Hence,
5.3 Copositivity of Q _{2}
Recall that Q _{2}=Q+L _{1}.
Proposition 1
In particular, Q _{2} is copositive in the cone , i.e., for every , Q _{2}(s)≥0 and equality happens if and if only s=0.
Proof.
 (a)
This follows immediately from Lemma 8(a), noting that L _{1}(s+s ^{′})=L _{1}(s)+L _{1}(s ^{′}). (b) The second inequality of (32) follows immediately from the definition.
In particular, Q _{2}(s) is an integer. By Lemmas 7(b) and 8(b), we have L _{1}(s)+Q(s)≥0, and equality happens only when L _{1}(s)=Q(s)=0. However, if L _{1}(s)=0, then by Lemma 7(b), N _{ γ,A }=2, and then Q(s)=N _{ γ,A }/2=1>0. Thus, we have proved that if s is an elementary centered state, then Q(s)>0. Since , we have Q(s)≥1.
Remark 15
In general, . In the proof, we show that for any elementary centered state s. One can also show that for all . This can be deduced from the fact that , see the discussion on fractional powers of J _{ K,n } in [18].
5.4 The lowest degree of the colored Jones polynomial
In the next proposition K is an alternating link.
Proposition 2
Proof.
 (a)
By (25), the minimal degree of q in w(s) is P _{1}(n)+Q _{2}(s). When s≠0, Proposition 1 implies that Q _{2}(s)>0. Hence the smallest degree of J _{ K,n }(q) is P _{1}(n). From the values of P _{1}(X,s) in Table 3 we see that . (b) follows easily from part (a) and (27).
The value of P _{0} in Table 4, Equation 25, and Proposition 2 imply the following:
Corollary 3
Remark 16
The minimal degree of the colored Jones polynomial J _{ K,n }(q) of an alternating link had been calculated using the Kauffman bracket skein module, and is given by , where s _{ A } is the number of circles obtained from by doing Aresolution at every vertex; see [16, Proposition 2.1] and keep in mind that the framing of K in [16] is different from the one in the current paper. Our result implies that P _{1}(n)=P1′(n). We will give a direct proof of this identity in the Appendix. Note also that s _{ A }−c _{+}=σ+1 (see [19, 50]), where σ is the signature of the link. Hence the lowest degree of q is given by .
6 From to the dual graph D ^{∗}
Recall from Section 1.4 that D ^{∗} is the dual graph of , considered as an unoriented graph. We have defined the Tait graph , the lattice Λ _{0} with its subsets Adm, Adm(n), and functions L,Q on Λ _{0}. Note that one does not need to bring D to a downward position by twisting in small neighborhood of crossing points in order to construct D ^{∗}.
where is the dual edge of e. Then is a linear map.
Proposition 3
Proof.
has one and exactly one solution λ∈Λ _{0}. This will prove the bijectivity of τ.
With the basis of Λ _{0}, every λ∈Λ _{0} has a unique presentation with k _{ v }=0 for v=v _{ ∞ }. We need to solve for k _{ v },v∈b from Eq. 38.
If k _{ v } is known, and v ^{′} is connected to v by an edge, then there is only one possible value for , namely . We call such the extension of the value k _{ v } at v along the edge e ^{∗}. Since the graph D ^{∗} is connected, and , we see that there is at most one solution λ∈Λ _{0} of (38).
Now let us look at the existence of solution of (38). Given , , and a path α of the graph Δ ^{∗} connecting v to , there is only one way to extend k _{ v }=a at v to v ^{′} along the path α. Denote by λ _{ α,a }(v ^{′}) the value at v ^{′} of this extension. When α is a closed path, i.e. v ^{′}=v, let Δ(α,a)=λ _{ α,a }(v ^{′})−a. We will show that Δ(α,a)=0 for any closed path α. This will prove the existence of the solution.
On , the closed path α encloses a region R. When the region is just a polygon of D ^{∗} (which must be a quadrilateral), the fact that Δ(α,a)=0 follows easily from (16). For general closed path α, since Δ(α,a) is the sum of , where α _{ j }’s are the boundaries of all the polygons of D ^{∗} in R, we also have Δ(α,a)=0.
The above fact shows that if we begin with , we can uniquely extend k _{ v } to all vertices of D ^{∗}, and obtain in this way an inverse of s.
 (b)
Because τ(λ)(e)=λ(e ^{∗}), (b) follows easily from the definitions.
 (c)
To prove (35), it is enough to consider the case , a basis vector. Let be the dual polygon. From the definition we have τ(v)=s _{ p }, where s _{ p } is the elementary centered state with support being the boundary of p. Now the identity L _{1}(τ(v))=L(v) follows from the value of L _{1} given in Lemma 7 and the definition of L. Actually, the definition of L was built so that (35) holds.
 (d)
We only need to check (37) for s=τ(v),v∈b. Let be the dual polygon. We already saw that τ(v)=s _{ p }. From the definition of L _{0} given by Table 3, we have that L _{0}(s _{ p }) is the number of negative vertices of p. Here a vertex is negative if it is negative as a crossing of the link diagram D.
There are two cases:
Case 1: p is a Bpolygon. Suppose or is an arbitrary orientation on edges of p. A vertex v of p is orincompatible if the orientations of the two edges incident to v are incompatible, i.e. the two incident edges are both going out from v or both coming in to v. Let f(or) be the number of all orincompatible vertices. It is easy to see that if or^{′} is obtained from or by changing the orientation at exactly one edge, then f(or)=f(or^{′}) (mod 2). It follows that f(or)=0 (mod 2) for any orientation or, since if we orient all the edges counterclockwise then f=0.
Let the orientation of D on the edges of p be denoted by or_{ D }. By inspection of Equation 15 one sees that a vertex v of p is a negative crossing if and only if v is or_{ D }incompatible. Thus, L _{0}(s _{ p })=f(or_{ D }), which is even by the above argument. On the other hand, 2L _{1}(s _{ p })=2 by Lemma 7.
Case 2: p is an Apolygon. By inspection of Equation 15, one sees that a vertex v of p is a positive crossing if and only if v is or_{ D }incompatible. This means L _{0}(s _{ p })= deg(v)−f(or_{ D })≡ deg(v) (mod 2), where deg(v) is the number of vertices of p, which is equal to the degree of v in the graph D ^{∗}. By Lemma 7, 2L _{1}(s _{ p })=−2+ deg(v). Hence, we also have (37).
Example 1
The reader can vefiry the equality (36).
Corollary 4
The dimension of (or ) is c _{ D }+ℓ, where ℓ is the number of connected components of the graph D.
Remark 17
One can show that the integervalued admissible colorings of D ^{∗} are the lattice points in a 2c _{ D } dimensional cone with 2c _{ D } independent rays.
7 Zero stability
In this section, we give a proof of the zero stability of the colored Jones polynomial of an alternating link and Theorem 2, which describes the zero limit as a generalized Nahm sum.
7.1 Expansion of Fand adequate series
Definition 4
Lemma 9
 (a)
For every , the set of xadequate series of order ≤t is a subring of . (b) If G(x,q) is xadequate of order ≤t, then it is xadequate of order ≤t ^{′} for any t ^{′}≥t. (c) If G(x,q) is xadequate of order ≤t, then the series f _{ n }(q)=G(q ^{ n },q) converges in the qadic topology and defines an element in for every n>t. (d) The sequence in (c) is stable and its associated series F _{ f }(x,q) satisfies F _{ f }(x,q)=G(x,q).
Proof.
Parts (a), (b) and (c) follow easily from the definition of an xadequate series.
which implies (d).
The negative powers of q in a _{ m }(q,s) come from the negative powers q ^{−s(v)},q ^{−s(e)} that appear in the expression of . Since s≥ max(s(v),s(e)), we have the following:
Lemma 10
7.2 Proof of zero stability
By part (b) of Proposition 1, Q _{2}(s)≥s. Then (45) implies that , and hence, the second sum on the righthand side of (47) is in .
By Lemma 10, a _{ m }(q,s) q ^{ ms} has only nonnegative powers of q. It follows that the righthand side of (48) belongs to . This completes the proof of Equation 46. □
Remark 18
Equation 46 is stronger than zero stability, and implies that for every , the coefficient of q ^{ m } in is independent of n for all n>m.
7.3 End of the proof of Theorem 2
To complete the proof of Theorem 2, it remains to prove that the righthand side of (44) is equal to that of (5). This follows from Proposition 3. □
Remark 19
The fact that D is reduced is used only in the proof of Lemma 7. As seen in Remark 14, Lemma 7 holds if D is Aadequate; hence, Theorem 2 holds if D is not necessarily reduced, but Aadequate.
7.4 Proof of Corollary 1
Fix a complex number q with q=a<1. We only need to show that the sum on the righthand side of (44) is absolutely convergent.
where g(m) is the number of such that Q _{2}(s)=m. Because Q _{2}(s) is quadratic and copositive in , g(m) is bounded above by a quadratic function of m for large enough m. From Equation 49, it follows that the righthand side of (52) is absolutely convergent. This completes the proof of Corollary 1.
8 Linearly bounded states
In this section, we will introduce a partition of the set of linearly bounded centered states, which will be key to the kstability of the colored Jones polynomial. Throughout this section, we fix a reduced, alternating, Ainfinite downward alternating link diagram D with c _{ D } crossings. Let . Recall that for a polygon , s _{ p } is the elementary centered state which support the boundary of p.
If were positive definite, it would be easy to prove the stability of . Unfortunately, Q is not positive definite, and the summation cone always contains directions where Q _{2}(s)=Q(s)+L _{1}(s) grows linearly, and not quadratically. For instance, if p is a Bpolygon, then Q _{2}(n s _{ p })=n is a linear function of n.
Definition 5
For a subset let denote the set of kbounded centered states in .
8.1 Balanced states at Bpolygons
Definition 6
We will say that a state is balanced at a Bpolygon p if s(v)≤s(p) for every vertex v of p, and equality holds for at least one vertex.
8.2 Seeds
In this section, we introduce seeds, their partial ordering, and relative seeds.
We say that two Bpolygons are disjoint if they do not have a common vertex. Suppose Π is a collection of disjoint Bpolygons. Let nbd(Π) be the set of all edges of incident to a vertex of a polygon in Π. Observe that every edge of a polygon in Π is in nbd(Π).
Definition 7
 (a)
A seed θ=(Π,σ) consists of a collection Π of disjoint Bpolygons and a map such that σ can be extended to a centered state which is balanced at every polygon in Π. Such s is called an extension of σ, and the set of all extensions of σ is denoted by . If σ is a seed and s an extension of it, then for all polygons p in Π and vertices v of p we can define σ(v)=s(v) and σ(p)=s(p) independent of s. In particular, σ is balanced at all polygons of Π. (b) The Bnorm  θ _{ B } of θ is the number of Bpolygons in Π.
Note that for the empty seed θ=∅, we have . Next, we define a partial order on the set of seeds.
Definition 8
Suppose θ=(Π,σ) and θ ^{′}=(Π ^{′},σ ^{′}) are seeds. Then θ≤θ ^{′} if Π⊂Π ^{′} and σ is the restriction of σ ^{′}. We write θ<θ ^{′} if θ≤θ ^{′} and θ≠θ ^{′}.
Observe that ∅≤θ for any seed θ. Moreover, if θ<θ ^{′}, then  θ _{ B }< θ ^{′} _{ B }. Since the number of Bpolygons is finite, we have the following simple but important fact:
Lemma 11
Every strictly increasing sequence of seeds is finite.
We now introduce relative seeds.
Definition 9
8.3 A partition of the set of kbounded states
In this section, we give a partition of the set of kbounded states and more generally, the set of kbounded states with seed θ. The next proposition will be proven in Section 10.
Proposition 4
Example 2
where m=s>s ^{′}, P is a Bpolygon and the support of s ^{′} is disjoint from P.
When θ is maximal, Proposition 4 implies the following:
Corollary 5
For every nonnegative integer k and every maximal seed θ, is a finite set.
The next proposition will also be proven in Section 10.
Proposition 5
 (a)
 (b)
Fixing θ<θ ^{′}, the presentation of given by Equation 54, with (m,s ^{′}) satisfying (55) is unique. In other words, the map (m,s ^{′})↦s given by (54) is a bijection between the set of pairs (m,s ^{′}) satisfying (55) and .
8.4 The weight of kbounded states
In this section, we express F(x,q,s) in terms of F(x,q,s ^{′}) for centered states s,s ^{′} related by Equation 54.
Definition 10
for every i,j≥0.
The next lemma is elementary.
Lemma 12
 (a)
If G(x,y,q) is weakly xadequate of order ≤t, then for every k≥t+1, l≥0. (b) If is weakly xadequate of order ≤t, then for every , is xadequate of order t. (c) The set of weakly xadequate series of order less ≤t is closed under addition and multiplication, i.e. it is a subalgebra of . (d) If G(x,y,q) is weakly xadequate of order ≤t, then it is weakly xadequate of order ≤t ^{′} for every t ^{′}≥t.
The next lemma uses the notation of Definition 9.
Lemma 13
Proof.
Here e is an edge and v is a vertex of a Bpolygon p in θ ^{′}∖θ, and is defined as in Section 8.1. Besides, in (58), v is the ending vertex of the edge e. Besides, each of is bounded from above by θ ^{′}∖θ, by definition.
where the second identity follows from a simplification of qfactorial using relations (58) to (61). Let us look at the factors in square brackets.
Since , the first square bracket factor is xadequate with order ≤θ ^{′}∖θ.
The second square bracket factor is in , where C=θ ^{′}∖θ(θ ^{′}∖θ+1)/2.
The third square bracket factor is a polynomial in x,y with coefficients in , and it is xadequate with order ≤θ ^{′}∖θ.
8.5 Stability away from the region of linear growth
In this section, we show the stability for the kunbounded centered states.
Proposition 6
is kstable.
Proof.
This completes the proof of the proposition.
9 Stability in the region of linear growth
Theorem 7.
is stable.
Remark 20
In particular, the above theorem holds when θ=∅, l=0 and G=1. In that case, Proposition 2 implies that and we conclude the stability of the colored Jones polynomial of an alternating link K.
Proof.
is kstable. We proceed by downwards induction, starting from the case when θ is maximal. This case follows from Corollary 5, which states that is a finite set, and Lemma 14
Assume that the statement holds for all θ ^{′} strictly greater than θ. We will show that the statement holds for θ. Then Lemma 11 implies that the statement holds for any seed θ.
where , and . G(x,y,q) is weakly xadequate of order ≤θ+l and θ+l≤θ ^{′}+l. Moreover, is weakly xadequate of order ≤θ ^{′}∖θ and θ ^{′}∖θ≤θ ^{′}≤θ ^{′}+l. Lemma 12 implies that G ^{′}(x,y,q) is weakly xadequate of order ≤θ ^{′}+l.
where the second identity follows from (63) and the abovementioned parametrization of , the third identity follows by changing notation s ^{′} to s and exchanging the two summations, and the fourth identity follows from Lemma 15 below, with G ^{″}(x,y,q) a weakly xadequate series of order ≤θ ^{′}+l. By induction hypothesis, the last sum of the above identity is kstable. This completes the proof of Theorem 7.
Lemma 14.
For a fixed , and G(x,y,q) weakly xadequate of order ≤t, the sequence is stable.
Proof.
Lemma 12 implies that q ^{−C } G(x,q ^{s},q) is xadequate and part (a) of Lemma 9 implies that q ^{−C } F(x,q,s)G(x,q ^{s},q) is xadequate, too. The result follows from part (d) of Lemma 9.
The next lemma is reminiscent to the notion of a qLaplace transform.
Lemma 15
Proof.
then (64) holds. It is easy to see that H is weakly xadequate of order ≤l+t.
10 Partition of the set of kbounded states
In this section, we will prove Propositions 4 and 5. We will fix an Ainfinite alternating, diagram D with c _{ D } crossings. We assume that D represent a nontrivial link, hence c _{ D }≥2.
10.1 Some lemmas regarding kcentered states
Note that if we think of s as a flow on , then a Bpolygon p is oriented counterclockwise and s(p) measures the amount that flows towards p. Equation 66 states that s(p) also equals to the amount that flows away from p. The next lemma motivates the definition of s(p).
Lemma 16
Proof.
where the sum is over all angles α of type A, and a and b are the svalues of the two edges forming the angle α.
Lemma 17
Suppose p is a Bpolygon, s a centered state, and . Then for s(e)≥m−s(p) for every .
Proof.
10.2 A decomposition of kbounded states
Definition 11
For a centered state s and a positive integer k, a polygon is (k,s)big if s achieves the maximal value s at one of the the vertices of p and s(p)<k.
It is obvious that every centered state s has some Bpolygons such that s achieves the maximum value at a vertex of those polygons. On the other hand, is it not true that every state s has (k,s)big polygons (for some k), which are always Bpolygons and always disjoint. However, this is true for kbounded states. This is the content of the following lemma. Its proof reveals a close connection between the notions of a kbounded state (given in Definition 5) and balanced polygons of Btype (given in Definition 6).
Lemma 18
Proof.
 (a)
Hence, s<8(k+1/3), which contradicts (71).
 (b)
To prove part (b), we first prove a few claims.
Claim 1. Suppose p is a Bpolygon of . Assume that s(e)>s−4k c _{ D } for an edge e of p. Then s(e ^{′})≥s(e)−2k for any edge e ^{′} of p incident to e.
Proof.
where the last inequality follows from (71). The above inequality contradicts the assumption that s is kbounded.
Let v be a vertex of where s(v)=s. One of the four edges incident to v, say e, has svalue ≥s/2. Let p be the unique Bpolygon of having e as an edge on the boundary. We will prove that p is (k,s)big. For this, we need to show that s(p)<k.
Claim 2. The svalue of every edge of p is at least s−4k c _{ D }.
Proof.
Besides e, suppose e ^{′} is the other edge incident to v which is also an edge of p, and f,f ^{′} are the other two edges which are not edges of p, as in Figure 8. Note that the number of edges of p is less than 2c _{ D }, the total number of edges of .
It follows that s(f ^{′})≤4(k+1/3)≤4k+2, and hence, by Equation (16), we have s(e ^{′})≥s−4k−2.
In the last inequality, we used the fact that k≥1 and c _{ D }≥2.
which contradicts the kboundedness of s. This completes the proof of Lemma 18.
10.3 Proof of Proposition 5
with and s ^{′}<m. The s ^{′}(e)=s(e) for every edge e outside Π ^{′}∖Π. Hence, if v is not a vertex of any p∈(Π ^{′}∖Π), then s(v)=s ^{′}(v)<m.
where the inequality follows from the fact that s ^{′} is balanced at p. But there is a vertex of p such that s ^{′}(v)=s ^{′}(p), and for which s(v)=m. It follows that the maximum of s(v) is m, or s=m.
Identity (56) follows right away from Lemma 16. Identity (57) follows that the fact that L _{0} is a linear map, L _{0}(s _{ p })=≡2L _{1}(s _{ p })≡2 (mod 2), by Lemmas 3 and 7.
Part(b). We have to show that s ^{′} and m are uniquely determined by s. In fact, by part (a), m=s. Then (74) shows that s ^{′} is determined by s and m. This completes the proof of Proposition 5. □
10.4 Proof of Proposition 4
Will show that if satisfying the lower bound (75), then there is a unique θ ^{′}>θ with θ ^{′}∖θ<k such that . This will prove Proposition 4.
Uniqueness. Assume that with θ ^{′}∖θ<k. Then s has a presentation given by (74). By Proposition 5(a), m=s is uniquely determined by s. In Section 10.3, we showed that if p∈(Π ^{′}∖Π) then there is a vertex v of p such that s(v)=s. We also have that s(p)≤θ ^{′}∖θ<k. Thus, every p∈Π ^{′}∖Π is (k,s)big.
Conversely, suppose p is a (k,s)big polygon. Then there is a vertex v of p such that s(v)=s. The proof of Proposition 5(a) showed that v is a vertex of a polygon p ^{′}∈Π ^{′}∖Π. Both p and p ^{′} are incident to v and both are (k,s)big. By Proposition 18(a), p=p ^{′}.
Thus, Π ^{′}∖Π is the set of all (k,s)big polygons. This determines Π ^{′} uniquely. Then (74) shows that s ^{′} is uniquely determined by s, and hence σ ^{′}, which is the restriction of s ^{′} on nbd(Π ^{′}), is uniquely determined by s. This completes the proof of uniqueness.
Existence. The proof of the uniqueness already shows us how to construct a presentation (74) for .
Let Ψ be the set of all (k,s)big polygons. If p is (k,s)big, then by Lemma 17 and (75), s(v)>s−k>θ for every vertex . This implies if p is disjoint from any polygon in Π. In particular, Π∩Ψ=∅. Let Π ^{′}=Π∪Ψ.
takes nonnegative integer value at every edge of , and hence is a centered state. Note that s(p)=s ^{′}(p) for any p∈Ψ since s and s ^{′} agree on any edge outside Ψ. We will show that (76) gives us the presentation (74).
If v is any vertex of for which s(v)=s, then Lemma 18(b) shows that v is a vertex of some polygon p∈Ψ. Hence, s ^{′}(v)=s(v)−(s−s(p))<s(v). This means s ^{′}<s.
On the other hand, if v is a vertex of p∈Ψ for which s(v)=s, then the above identity shows that s ^{′}(v)=s ^{′}(p). This means s ^{′} is balanced at every p∈Ψ. Since s ^{′}=σ in nbd(Π), it is balanced at every p∈Π. Thus, s ^{′} is balanced at every p∈Π ^{′}=Π∪Ψ.
Let σ ^{′} be the restriction of s ^{′} on nbd(Π ^{′}) and θ ^{′}=(Π ^{′},σ ^{′}). Then , and (76) gives us the presentation (74), and we have .
Thus, we conclude that every satisfying (75) is an element of for some θ ^{′}>θ with θ ^{′}∖θ<k. This concludes the proof of the existence, and whence Proposition 4. □
11 Proof of Theorem 4
In this section, we prove Theorem 4. It is wellknown that pointwise sums and products of qholonomic sequences are qholonomic (see [26, 27]). Moreover, the colored Jones polynomial (J _{ K,n }(q)) of every link is qholonomic [52]. Using (34), we deduce that is qholonomic for every alternating link K. Using a recursion relation (7) for and the stability Theorem 1, and collecting powers of q and q ^{ n }, it follows that Φ _{ K,k }(q) is qholonomic.
Using a linear recursion for Φ _{ K,k }(q), it is easy to see that mindeg_{ q }(Φ _{ K,k }(q)) is bounded below by a quadratic function of k; see for example [53, Thm.10.3]. A stronger statement is known [4], namely mindeg_{ q }(Φ _{ K,k }(q)) is a quadratic quasipolynomial of k. This proves Equation 8.
Equation 9 follows from Equation 8 using Lemma 19 below. This concludes the proof of Theorem 4. □
Lemma 19
Assume that for all k. Then the following are equivalent:

mindeg_{ q }(Φ _{ k }(q))≥−C _{1} k ^{2}−C _{2} for all k.

mindeg_{ q }(R _{ k,n }(q))≥n+1−C _{1}(k+1)^{2}−C _{2} for all k and all n large enough.
Proof.
Now, (a) implies (b) by Equations 77 and 80 and (b) implies (a) by Equations 77 and 79.
12 An algorithm for the computation of Φ _{ K,k }(q)
12.1 A parametrization of onebounded states
In this section, we will compute explicitly the series Φ _{ K,1}(q) of an alternating knot in terms of a planar projection as in Theorems 2 and 3. We begin with a corollary of Proposition 4 for k=1 and θ=∅. See also Example 2.
Corollary 6
and s(e)=0 if e is an edge of which contains a vertex of p. Moreover, (P,s ^{′}) are uniquely determined by s.
12.2 The computation of Φ _{ K,1}(q) in terms of a planar diagram
where and are the vertices and the edges of .
Using Section 6, we can convert the above formula for Φ _{ K,1}(q) in terms of the Tait graph of an alternating planar projection of K. This concludes the proof of Theorem 3. □
The above algorithm can be used to give a formula for Φ _{ K,k }(q) as follows. Separate the statesum of Equation 33 in two regions:

s is not kbounded.

s is kbounded.
Observe that t≤k. If t=k we stop. Else, replace (s,k) by (s ^{′},k−t) in the above step and and run it again. Keep going. Since each step requires at least one new polygon of Btype which is vertexdisjoint from the previous ones, this algorithm terminates in finitely many steps.
Remark 21
Using the parametrization of twobounded states from Example 2 and the above algorithm, the reader may obtain a formula for Φ _{ K,2}(q).
13 Φ _{0}is determined by the reduced Tait graph
In this section, we prove Corollary 2. Throughout, we use the following convention on graphs: a graph is a finite onedimensional CWcomplex without loop edge.
Recall that a plane graph is a pair γ=(Γ,f), where Γ is a finite connected planar graph and is an embedding. For example, if D is an alternating nonsplit link diagram, then the Tait graph is a plane graph. One can recover K from up to orientation.
13.1 From plane graph to nonoriented alternating link
Note that D(γ) is a nonoriented alternating Ainfinite link diagram. The resulting D(γ), although alternating, may be reducible. If D is an alternating link diagram, and be its Tait graph, then .
Exercise 1
Show that D(γ) is reducible if and only if Γ contains a cut edge, i.e. an edge e such that removing in interior of e make Γ disconnected.
where Φ _{ γ,0} is given by the the righthand side of (5) with .
The dual D ^{∗}(γ) (in S ^{2}) of D(γ) can be constructed directly from γ as follows: in each region p of γ choose a point u _{ p } and connect u _{ p } to all the vertices of p by edges inside the region p so that the edges do not intersect except at u _{ p }. Then D ^{∗}(γ) is the plane graph whose vertex set is and whose edges are all the edges just constructed. The edges of γ are not edges of D ^{∗}.
13.2 kedgeconnected graphs
Recall that a vertex v of a graph Γ is a cut vertex if Γ is the union of two proper subgraphs Γ _{1} and Γ _{2} so that Γ _{1}∩Γ _{2}={v}. A graph is twoconnected if it is connected and has no cut vertex.
A pair (u,v) of vertices of Γ is a cut pair if Γ is the union of two proper subgraphs Γ _{1} and Γ _{2}, neither of which is an edge, so that Γ _{1}∩Γ _{2}={u,v}.
By Whitney’s theorem [54], two planar embeddings of a twoconnected planar graph are related by a sequence of Whitney flips, composed with a homeomorphism of S ^{2}. Since, Conway mutation does not change the colored Jones polynomial [55], from (82) we have the following:
Lemma 20
If γ _{1} and γ _{2} are two planar embeddings of the same twoconnected graph, then .
13.3 Planar collapsing of a bigon
Suppose γ=(Γ,f) is a plane graph, and among the regions of there is a bigon u with vertices v _{1},v _{2} and edges e _{1},e _{2}. Let β be the plane graph obtained from γ by squeezing the bigon into one edge, called e, so that the bigon disappears and both e _{1} and e _{2} becomes e. We call γ→β a planar collapsing.
Lemma 21
Proof.
Here a,b _{1},b _{2} are the coordinates of λ at respectively v _{ u },v _{1},v _{2}; and b _{1} and b _{2} are fixed in the sum. By the wellknown Durfee’s identity (see [56, Eq.2.6]), the factor in the square bracket is equal to 1.
Remark 22
Suppose K _{1} and K _{2} are alternating links such that after several planar collapsings from and and one gets the same plane graph, then the above lemma says that . This was proved in [24] by another method.
13.4 Abstract collapsing
Suppose Γ _{1} is an abstract graph with a pair of parallel edges e _{1},e _{2}. Removing the interior of e _{1}, from Γ _{1} we get a graph Γ _{2}. We say that the move Γ _{1}→Γ _{1} a collapsing. Note that if Γ _{1} is twoconnected then Γ _{2} is also twoconnected.
Lemma 22
Suppose γ _{1}=(Γ _{1},f _{1}) is a plane graph, and Γ _{2} is obtained from Γ _{1} by collapsing a pair of parallel edges e _{1},e _{2}. Then there is a planar embedding γ _{2} of Γ _{2}, γ _{2}=(Γ _{2},f _{2}), such that .
Proof.
In the planar embedding , e _{1} and e _{2} bound a region which may contain a subgraph Γ _{0} of Γ _{1}. Note that the common vertices v _{1} and v _{2} of e and e _{2} form a cut pair for Γ _{1}. By flipping f _{1}(e _{2})∪f _{2}(Γ _{0}) through v _{1} and v _{2}, from γ _{1} we get a new plane graph γ _{3}=(Γ _{1},f _{3}) in which f _{3}(e _{1}) and f _{3}(e _{2}) form a bigon, and the result of planar collapsing this bigon is denoted by γ _{2}. By Lemmas 21 and 20, we have .
13.5 Proof of Corollary 2
We will first prove the following statement:
Lemma 23
Suppose γ _{ i }=(Γ _{ i },f _{ i }) for i=1,2 are twoconnected graphs such that Γ1′=Γ2′ as abstract graphs. Then .
Proof.
Case 1: Both Γ _{1} and Γ _{2} do not have multiple edges. Then Γ i′=Γ _{ i }, hence Γ _{1}=Γ _{2}, and γ _{1} and γ _{2} are planar embeddings of the same twoconnect graph. Lemma 20 tells us that .
Case 2: General case. This case is reduced to Case 1 by induction on the number of total pairs of parallel edges in Γ _{1} and Γ _{2}. If there is no pair of parallel edges, this is Case 1. Suppose Γ _{1} has a pair of parallel edges, and let Γ _{3} be the result of abstract collapsing this pair of parallel edges. By Lemma 22, there is a planar embedding γ _{3} of Γ _{3} such that . Note that Γ3′=Γ1′=Γ2′. By induction, we have . This proves .
Let us proceed to the proof of Corollary 2. Suppose K _{1} and K _{2} are alternating links such that is isomorphic to as abstract graphs. We can assume that both K _{1} and K _{2} are nonsplit. For i=1,2, let be the plane Tait graph of a reduced Ainfinite alternating link diagram of K _{ i }. Note that is connected since K _{ i } is nonsplit. Moreover does not have a cut vertex since D _{ i } is reduced. That is, is twoconnected. From Lemma 23 and Theorem 2, we have . This completes the proof of Corollary 2.
14 Examples
Twist knots
Twist knot  K _{−4}  K _{−3}  K _{−2}  K _{−1}  K _{1}  K _{2}  K _{3}  K _{4} 

Rolfsen notation  10_{1}  8_{1}  6_{1}  4_{1}  3_{1}  5_{2}  7_{2}  9_{2} 
Recall that sgn(n)=+1,0,−1 when n<0,n=0,n>0 respectively.
Theorem 8.
Equation 83 implies that for p<0, are modular forms [58]. On the other hand, when p>1, is not modular of any weight, according to K. Ono. This disproves any conjectured modularity properties of Φ _{0}(q), even for 5_{2}. On the other hand, is a false theta series of Rogers.
for all natural numbers b. It was pointed out to us by D. Zagier that the above identity follows immediately from the Jacobi triple product identity, discussed in detail in [58].
Appendix
Proof of the statesum formula for the colored Jones function
In this section, we give a proof of Equation 14 which we could not find in the literature. We begin by recalling the definition of the colored Jones polynomial using Rmatrix.
Link invariant associated to a ribbon algebra
Quantum link invariants can be defined using a ribbon Hopf algebra. We recall the formula for the invariant here. For further details, see [59] or [14].
where S is the antipode of the .
Suppose V is a module, and K is a framed link with a downward planar diagram D, where the framing is the blackboard framing. The dual space V ^{∗} has a natural structure of a module. Fix a basis {e _{ j }} of V and a dual basis of V ^{∗}.
The quantum invariant is defined through tangle operator invariants as follows.
The case
The colored Jones polynomial is the quantum link invariant corresponding to the ribbon Hopf algebra , the quantized enveloping algebra of . There are two versions of in the literature; we will use here the version used in [14, 60], which has the opposite coproduct structure of the one used in [61, 62]. The ground ring is not a field; but the theory carries over without changes.
where D= exp(h H⊗H/4), which is called the diagonal part.
is known as the canonical basis of V _{ n }.
For a framed link K, let be the invariant of K with color V _{ n }. It is known that if one increases the framing of a component of K by one, then J _{ K,n } gets multiplied by .
Define in the same way as in the definition of , with b replaced by and b ^{−1} replaced by . Then is an invariant of unframed links. Since for some , when dividing by the smallest monomial, both and are the same.
Rmatrix in the canonical basis
where J _{ K,n } is the given in Section 2.4.
The lowest degree of the colored Jones polynomial of an alternating link
We fix an Ainfinite, reduced, alternating, downward diagram D of a link K. Corollary 2 shows that the minimal degree of the colored Jones polynomial J _{ K,n } is given by , where the sum is over all the local extreme points of D. On the other hand, the Kauffman bracket approach gives the minimal degree as , see [19] and also [3, 16]. Here we show that the two results agree. Recall that s _{ A } is the number of circles obtained from D after doing Asmoothenings at every vertex crossing of D. If D is a connected graph, then s _{ A } is the number of Avertices of Δ ^{∗}.
Lemma 24
Remark 23
we have c _{−}=0, s _{+}=2, and there are two clockwise local extrema and two counterclockwise local extrema giving .
Proof.
(of Lemma 24) Suppose D ^{′} is the result of doing Atype resolution at every crossing of D. Since all crossings of D are downward, an Atype resolution at a positive crossing of creates no local extreme point, while an Atype resolution at a negative crossing created two local extreme points, each has winding weight 1/2.
By Lemma 2(b), D ^{′} consists of s _{ A } circles, each having winding number 1. Hence , and we get .
Regularity of Nahm sums
where f _{ i } are homogeneous polynomials of degree i and f _{ d } not identically zero. Assume that . Let , and .
Proposition 7
The following are equivalent:
Proof.
 (a)
⇒ (b) Fix . By properness, it follows that the sequence f(n s) is unbounded thus it has a subsequence that goes to infinity. Suppose that f _{ i }(s)=0 for i<i _{0} and . Then, goes to infinity. It follows that .
 (b)
⇒ (c) Indeed, (b) implies that for all s∈X we have f _{ d }(s)≥0, and if f _{ d }(s)=0 then (without loss of generality) we assume that f _{ d−1}(s)≥0. Since f _{ i } is homogeneous, it follows that for all we have f _{ d }(s)≥0, and if f _{ d }(s)=0 then f _{ d−1}(s)≥0. Since f _{ i } is continuous, it follows that for all , we have f _{ d }(s)≥0, and if f _{ d }(s)=0 then f _{ d−1}(s)≥0. Let S denote the unit sphere in the Euclidean space with the Euclidean norm ·_{2}. Then, f _{ d }:[0,∞)^{ r }∩S→[0,∞), and if Z _{ d }={x∈[0,∞)^{ r }∩S  f _{ d }(s)=0}, then f _{ d−1}(Z _{ d })>0. By continuity, choose an open neighborhood U _{ d } of Z _{ d } in [0,∞)^{ r }∩S with closure such that . Then, f _{ d }(([0,∞)^{ r }∩S)∖Z _{ d })⊂(0,∞) and by compactness it follows that there exists c ^{′}>0 such that f _{ d }(([0,∞)^{ r }∩S)∖Z _{ d })∈[c ^{′},∞) and . It follows that (f _{ d }+f _{ d−1})([0,∞)^{ r }∩S)⊂[c ^{′},∞), and by homogeneity this implies that for all , we have . On the other hand, by homogeneity, we have for i<r−1. Since c ^{′} x ^{ r−1}−c ^{′′} x ^{ r−2}≥c x for some c>0 and for all x sufficiently large, (c) follows.
 (c)
⇒ (a) is immediate.
For example, the function Q _{2} in Section 5 is proper and bounded from below. Actually, Q _{2}≥0 on the cone .
Experimental formulas for knots with a low number of crossings
The formulas for Φ _{0}(q) presented below agree to the first 8 values with the KnotAtlas table of BarNatan, and also to 50 values with the Nahm sum formula of Theorem 1. The formulas are proven only for 3_{1} and 4_{1} knot, and remain conjectural for all others.
Zero limits of some knots
K  c _{−}  c _{+}  σ  Φ _{ K ,0} ( q )  

3_{1}=−K _{1}  3  0  2  h _{3}  1 
4_{1}=K _{−1}  2  2  0  h _{3}  h _{3} 
5_{1}  5  0  4  h _{5}  1 
5_{2}=K _{2}  0  5  −2  h _{3}  
6_{1}=K _{−2}  4  2  0  h _{3}  h _{5} 
6_{2}  4  2  2  h _{3}  
6_{3}  3  3  0  
7_{1}  7  0  6  h _{7}  1 
7_{2}=K _{3}  0  7  −2  h _{3}  
7_{3}  0  7  −4  h _{5}  
7_{4}  0  7  −2  h _{3}  
7_{5}  7  0  4  
7_{6}  5  2  2  
7_{7}  3  4  0  
8_{1}=K _{−3}  6  2  0  h _{3}  h _{7} 
8_{2}  6  2  4  h _{3}  
8_{3}  4  4  0  h _{5}  h _{5} 
8_{4}  4  4  2  h _{3}  
8_{5}  2  6  −4  h _{3}  ??? 
K _{ p },p>0  0  2p+1  −2  h _{3}  
K _{ p },p<0  2p  2  0  h _{3}  h _{2p+1} 
T(2,p),p>0  2p+1  0  2p  h _{2p+1}  1 
Let us summarize some observations:
Declarations
Acknowledgements
An early version of the paper was presented in talks of the first author to a Spring School in Geometry and Quantum Topology in the Diablerets 2011, and in the Mathematische Arbeitstagung in Bonn, 2011. The authors wish to thank the organizers of the above conferences for their hospitality, C. Armond and O. Dasbach for explaining to us their beautiful work. The firstnamed author wishes to thank T. Dimofte and D. Zagier for their interest and encouragement and for the generous sharing of their ideas. The authors were supported in part by NSF.
Authors’ Affiliations
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