Pretzel Knots and q-Series

Pretzel Knots and q-Series

Mohamed Elhamdadi Department of Mathematics, University of South Florida, Tampa, FL 33647 USA  and  Mustafa Hajij Department of Mathematics, University of South Florida, Tampa, FL 33647 USA
Abstract.

The tail of the colored Jones polynomial of an alternating link is a -series invariant whose first terms coincide with the first terms of the -th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In this paper, we study -series identities coming from the colored Jones polynomial of pretzel knots. We prove a false theta function identity that goes back to Ramanujan and we give a natural generalization of this identity using the tail of the colored Jones polynomial of Pretzel knots. Furthermore, we compute the tail for an infinite family of Pretzel knots and relate it to false theta function-type identities.

1. Introduction

The discovery of the Jones polynomial using Von Neumann algebras [J1, J2] and its generalizations [HOMFLY] and [PT] lead to quantum invariants of knots and -manifolds. The Kauffman bracket polynomial [Kauffman] is the simplest interpretation of the Jones polynomial using knot diagrams. Reshetikhin and Turaev [RT] gave the first rigorous construction of quantum invariants as linear sums of quantum invariants of framed links. Soon after, various approaches of constructing quantum invariants were developed using different methods such as using surgery along links [BHMV92, Lic92, TuraevWenzl93] and simplicial complexes [TuraevViro92].
One of the earliest connection between the colored Jones polynomial and Ramanujan type -series was made in [Hikami] in which the author investigated the asymptotic behaviors of the colored Jones polynomials of torus knots. However, the point of view in [Hikami] is different from the point of view of [Hajij1, Hajij2] that we shall adopt here. This point of view allows us to prove more -series identities in a structured manner. Among many interesting properties that the tail of the colored Jones polynomial enjoys as -series is that it is equal to theta functions or false theta functions for many knots with small crossing numbers. For instance all knots in the knots table up to , the tail of their colored Jones polynomial are Ramanujan theta, false theta functions or a product of these functions as demonstrated in [CodyOliver]. This does not seem to be the case of knot whose tail is computed in [Hajij1]. More interestingly, the study of the tail has been used to prove Andrews-Gordon identities for the two variable Ramanujan theta function in [CodyOliver] and a corresponding identities for the false theta function in [Hajij2]. These two families of -series identities were obtained from investigating -torus knots. For -series techniques proving these identities refer to [Robert].
In this paper we show that similar observations hold for other natural family of knots, namely Pretzel knots. In particular, we show that pretzel knots give rise to a natural family of -series identities. The paper is organized as follows. In section 2 we review the basics of skein theory, some number theory relevant to our work, and some review of the colored Jones polynomial. In section 3 we list the main results of this paper. Section 4 is devoted to Ramanujan type identities that were recovered in the literature using the tail of the colored Jones polynomial and we show how our contribution here fits in this literature. In section 5 we give an explicit formula for the tail of colored Jones polynomial of the Pretzel knots where . In section 6 we use two skein theoretic techniques to compute the tail of the colored Jones polynomial of a certain family of pretzel knots and we show that these computations give rise to a Ramanujan type identities.

2. Review of Skein Theory and Colored Trivalent Graphs

2.1. Skein Theory

Let denotes the set of rational functions where . Let be an orientable -manifold. A framed link in is a disjoint union of oriented annuli embedded into Let be the set of all isotopy classes of framed links in . We consider the empty link to be an element of . Denote by the free -module generated by . Three framed links , , and are said to be Kauffman skein related if they can be embedded in identically except in a ball where they appear as in the Figure 1 below.

If , and are skein related then an expression of the form is called a skein relation. On the other hand, an expression of the form

where consists of an element in and the trivial framed unknot, is called a weight relation. Let denote by the smallest submodule of that is generated by all possible skein relations and weight relations. The Kauffman bracket skein module of is defined to be the quotient module The definition of the Kauffman bracket skein module can be extended to include -manifolds with boundary. In this case we call the resulting module the relative Kauffman bracket skein module. More precisely, the definition goes as follows. Specify a finite set of marked points on the boundary of A band is a surface that is homeomorphic to . An element in the set is an isotopy class of an oriented surface embedded into and decomposed into a union of finite number of framed links and bands joining the designated boundary points. The relative Kauffman bracket skein module is the quotient module In this paper we are interested in the case when is where is an orientable surface. When this is the case, one thinks of a framed link in as a link diagram in with framings being determined by parallel curves in . The Kauffman bracket skein module of will be denoted by
In this paper we will use the Kauffman bracket skein module of . This module is isomophic to . To see this, let be a diagram in . Using the definition of the normalized Kauffman bracket we can write where is the empty link. This defines an isomorphism between and induced by sending to . The second module that we will use is the relative skein module of the disk with marked points on the top and points on the bottom. Two diagrams in can be concatenated to produce another diagram in This defines a multiplication on that makes this module an associative unital algebra over . This algebra is called the Temperley-Lieb algebra and is denoted usually by .

For each there exists a unique idempotent called the Jones-Wenzl idempotent. We will use a graphical notation for which is due to Lickorish [Lic92]. In this graphical notation one thinks of as an empty box with strands coming in the top and strands leaving the bottom. The Jones-Wenzl idempotent enjoys a recursive formula that is due to Wenzl [Wenzl]. The recursive formula can be stated graphically as follows :

 (2.1)

where

 Δn=(−1)nA2(n+1)−A−2(n+1)A2−A−2.

Furthermore, the idempotent has the following properties:

 (2.2)

And:

 (2.3)

If is a diagram in and is a diagram in then we define to be the diagram in obtained by joining the diagrams and as follows:

Before we introduce other skein modules we need the concept of wiring maps to deal with linear maps between skein modules.

2.1.1. Wiring Maps

We can relate various skein modules by linear maps induced from maps between surfaces. Let and be two oriented surfaces with marked points on their boundaries. A wiring is an orientation preserving embedding of into along with a fixed wiring diagram of arcs and curves in such that the boundary points of the arcs consists of all the marked points of and . Any diagram in induces a diagram in by extending by a wiring diagram. A wiring of into induces a module homomorphism

 S(W):S(F)⟷S(F′)

defined by for any diagram in . More details about skein wiring can be found in the paper of Morton [Morton]. The following is an example which induces a map between the Temperley-Lieb algebra and the Kauffman skein module of the -sphere.

Example 2.1.

Consider the square with marked points on the top edge and marked points on the bottom edge. Embed in and join the points on the top edge to the points on the bottom edge by parallel arcs as follows:

For each , this wiring induces a module homomorphism:

 trn:TLn⟶S(S2)

This map is usually called the Markov trace on .

2.2. Colored Trivalent Graphs

Now we will define certain submodules of the skein module of the disk with marked points on the boundary. These modules will be useful in the computation of the tail of the colored Jones polynomial. Consider the skein module of with specified points on the boundary. Partition the set of the points on the boundary of the disk into sets of , and points respectively and at each cluster of points we place an appropriate idempotent, i.e. the one whose color matches the cardinality of this cluster. The skein module constructed in this method will be denoted by The skein module is either zero dimensional or one dimensional. The skein module is one dimensional if and only if the element shown in Figure 2 exists. For this element to exist it is necessary to find non-negative integers and such that , and .

The following definition characterizes the existence of this skein element in terms of the integers , and .

Definition 2.2.

A triple of non-negative integers is admissible if is even and

When the triple is admissible, one can write , , and . In this case we will denote the skein element that generates the space by . Note that when the triple is not admissible then the space is zero dimensional. The fact that the inside colors are determined by the outside colors allows us to replace by a trivalent graph as follows:

Similarly, we define the module of the disk . Precisely the skein module is a submodule of the skein module of the disk with marked points on the boundary and we place the idempotents , , , and on the appropriate set of points as we did for . See Figure 4.

In order to perform our computation for the tail of the colored Jones polynomial it is important to understand the evaluation of certain skein elements in . Th evaluation of these skein elements can be understood as the evaluation of certain colored trivalent graphs in . A colored trivalent graph is a planer trivalent graph with edges labeled by non-negative integers. One usually uses the word color to refer to a label of the edge of a trivalent graph. A colored trivalent graph is called admissible if the three edges meeting at a vertex satisfy the admissibility condition of the definition 2.2. If is an admissible colored trivalent graph then the Kauffman bracket evaluation of is defined to be the evaluation of as an element in after replacing each edge colored by the projector and each admissible vertex colored by the skein element , as in Figure 3. If a colored trivalent graph has an inadmissible vertex then we will consider its evaluation in to be zero. We will need the evaluation of the following important colored trivalent graphs shown in Figure 5.

For an admissible triple , an explicit formula for the theta coefficient, denoted , was computed in [MV] and is given by:

 (2.4)

where and are the interior colors of the vertex . The tetrahedron coefficient is defined to be the evaluation of the graph appearing on the right handside of Figure 5 and a formula of it can be found in [MV]. The tetrahedron graph in Figure 5 is denoted by . The following identity holds in :

Recall that, for any integers such that , the quantum binomial coefficients are defined by :

 [li]q=(q;q)l(q;q)i(q;q)l−i.

where is -Pochhammer symbol which is defined as

 (a;q)n=n−1∏j=0(1−aqj).

We will need the following identity [Hajij1].

Theorem 2.3.

(The bubble expansion formula) Let , and ; . Then

 (2.6)

where

 ⌈mnkl⌉i:=(−A2)i(i−l)l−i−1∏j=0Δk−j−1i−1∏s=0Δn−s−1Δm−s−1l−1∏t=0Δn+k−t−1Δm+k−t−1[li]A4l−i−1∏j=0Δm+n+k−i−j.

We will denote the skein element on the right handside of (2.6) by and we will call it the bubble skein element.

2.3. The Tail of The Colored Jones Polynomial

We briefly review the basics of the head and the tail of the colored Jones polynomial. For more details see [Hajij1, Hajij2].

Let be a framed link in . Decorate every component of , according to its framing, by the Jones-Wenzl idempotent and consider the evaluation of the decorated framed link as an element of . Up to a power of , that depends on the framing of , the value of this element is defined to be the (unreduced) colored Jones polynomial . Recovering the reduced Jones polynomial is a matter of changing a variable and dividing by . Namely,

 Jn+1,L(q)=~Jn,L(A)Δn∣∣∣A=q1/4 (2.7)

If and are elements in , we write if their first coefficients agree up to a sign. It was proven in [CodyOliver] that the coefficients of the colored Jones polynomial of an alternating link stabilize in the following sense: For every , we have . This motivated the authors of [CodyOliver] to define the tail of the colored Jones polynomial of a link. More precisely, define the -series series associated with the colored Jones polynomial of an alternating link whose coefficient is the coefficient of . Stated differently, the tail of the colored Jones polynomial of a link is defined to be a series , that satisfies for all . In the same way, the head of the colored Jones polynomial of a link is defined to be the tail of . The head and the tail of the colored Jones polynomial of an alternating link can be recovered from a sequence of skein elements in . The study of this sequence of skein elements is relatively easier than the study of the entire colored Jones polynomial. For more details see [CodyOliver] and [Hajij2]. We recall this fact here. Let be a link in and be an alternating knot diagram of . Consider the all -smoothings state of , the state obtained by replacing each crossing by a -smoothing. We record the places of this smoothing by a dashed line as can be seen in Figure 6 for an example. Write for the all -smoothing state and consider the skein element obtained from by decorating each circle in with the Jones-Wenzl idempotent and replacing each dashed line in with the Jones-Wenzl idempotent. See Figure 6.

The following theorem from [CodyOliver] relates the tail of the colored Jones polynomial of an alternating link to the skein element .

Theorem 2.4.

Let be an alternating link in and let be an alternating diagram of . Then

 ~Jn,L(A)≐4(n+1)S(n)B(D)

This theorem states basically that the study of the tail of the colored Jones polynomial of the alternating knot can be reduced to the study of the tail of the sequence of skein elements . This theorem also implies that the tail of the colored Jones polynomial depends on the so called the reduced B-graph of the diagram . The -graph of the diagram , denoted is the graph whose vertices are the circles of and whose edges are the dashed lines. The reduced -graph of , denoted by , is obtained from by replacing parallel edges by a single edge. See the most right two drawings in Figure 6.

Remark 2.5.

Since the colored Jones polynomial of a diagram depends only on its reduced -graph, we will sometimes use the term the tail a graph to refer to the tail of colored Jones polynomial of an alternating knot diagram such that . Conversely, Given a planar graph , we can obtain an alternating knot diagram such that by replacing every edge in by a crossing as illustrated in Figure 7. For this reason, if is a planar graph then the tail of will be denoted by . Furthermore, the notation will refer to the skein element obtained from the reduced graph by replacing each vertex with a circle and each edge with the Jones-Wenzl projector.

Remark 2.6.

In general the computations of the tail of the colored Jones polynomial is done for the reduced case. In order to use Theorem 2.4 one needs to do a change of variable and normalize by as can be seen from the relation (2.7).

The tail of the colored Jones polynomial has been computed for all knots in the knot table up to the knot by Armond and Dasbach in [CodyOliver]. In [Hajij2], the second author gave a formula for .

3. Main Results

In this section we list the main results of the paper. Let be positive integers. Denote by the pretzel knot with crossing regions given in Figure 8.

In the following theorem, we give a formula for the tail of the colored Jones polynomial of the pretzel knot for .

Theorem 3.1.

The tail of the pretzel knot is given by

 TP(2k+1,2,2u+1)(q)=(q;q)2∞∞∑l1=0...∞∑lk=0∞∑p1=0...∞∑pu=0g(q;l1,...,lk)g(q;p1,...,pu)(q;q)i+j

where

 g(q;l1,...,lk)=qk∑j=1(ij(ij+1))(q;q)2lkk−1∏j=1(q;q)lj

with .

This formula generalizes the one of the tail of colored Jones polynomial of the knot given in [Hajij1]. Furthermore, we give a formula for the tail of the colored Jones polynomial of the pretzel knot with crossing regions.

Proposition 3.2.

Let and let denotes with crossing regions. Then

 TPk(q)=(q;q)k∞∞∑i=0qi(q;q)ki.

We use skein theoretic techniques to give another method to compute and we obtain the following identity.

Corollary 3.3.

For we have

 (q;q)∞∞∑i=0qi(q;q)k+1i=∞∑i1=0...∞∑ik=0q∑kj=1ij+i2j+∑ks=2∑kj=sis−1ij∏kj=1(q;q)ij(q;q)∑js=1is

This gives a natural generalization of the following well-known false theta function identity (The Lost Notebook and Other Unpublished Papers; page 200 in [Ramanujan]) :

 (q;q)2∞∞∑i=0qi(q;q)2i=(q;q)∞∞∑i=0qi2+i(q;q)2i.

4. Alternating Knots and Rogers-Ramanujan Type Identities

In this section, we review the Rogers-Ramanujan type identities that were recovered in the literature using techniques related to the tail of the colored Jones polynomial of alternating links. Furthermore, we show the false theta function type identities that we recover in this paper from Pretzel knots.

The general two variable Ramanujan false theta function is given by (e.g. [Andrews]):

 Ψ(a,b)=∞∑i=0ai(i+1)2bi(i−1)2−∞∑i=1ai(i−1)2bi(i+1)2 (4.1)

When and , we obtain the following well-known identities:

 Ψ(q3,q)=∞∑k=0(−1)kqk2+k2=(q;q)∞∞∑k=0qk2+k(q;q)2k=(q;q)2∞∞∑k=0qk(q;q)2k (4.2)

In [Hajij1], the second author recovered the second identity in 4.2 using the tail of the -torus link. Furthermore, the tail of the colored Jones polynomial of -torus links, where , to give a natural extension of the same identity 4.2. For all , this identity is given by:

 Ψ(q2k−1,q)=(q;q)∞∞∑l1=0∞∑l2=0...∞∑lk−1=0qk−1∑j=1(ij(ij+1))(q;q)2lk−1k−2∏j=1(q;q)lj (4.3)

where . On the other hand, a similar identity for the theta function, known as Roger-Ramanujan identity for the two-variable theta function, can be recovered from the tail of the colored Jones polynomial of -torus knots. Recall that the general two variable Ramanujan theta function is defined by:

 f(a,b)=∞∑i=0ai(i+1)/2bi(i−1)/2+∞∑i=1ai(i−1)/2bi(i+1)/2 (4.4)

The function specializes to:

 f(−q2k,−q)=∞∑i=0(−1)iqk(i2+i)qi(i−1)/2+∞∑i=1(−1)iqk(i2−i)qi(i+1)/2. (4.5)

For the Roger-Ramanujan identity for the theta function is given by:

 f(−q2k,−q)=(q;q)∞∞∑l1=0∞∑l2=0...∞∑lk−1=0qk−1∑j=1(ij(ij+1))k−1∏j=1(q;q)lj (4.6)

where . The identities 4.3 and 4.6 were recovered using a unified skein theoretic method in [Hajij1]. Note that the identities (4.6) and (4.3) are coming from cyclic graphs with odd and even number of vertices respectively. It is plausible to think that a natural family of knots, or graphs, gives rise to a natural family of -series identities. In this paper we recover the third identity (4.2) using the tail of the colored Jones polynomial. This -series correspond to the graph given in the following Figure.

Furthermore, we give a natural generalization of this identity using the tail of the graph , where , given in Figure 13. Note that the graph corresponds to the the pretzel knot in Proposition 6.2.

We show that this generalization is given by:

 (q;q)∞∞∑i=0qi(q;q)k+1i=∞∑i1=0...∞∑ik=0q∑kj=1ij+i2j+∑ks=2∑kj=sis−1ij∏kj=1(q;q)ij(q;q)∑js=1is (4.7)

5. The Tail of the Colored Jones Polynomial of the Pretzel Knots P(2k+1,2,2u+1)

In [Hajij1], the second author computed the tail of the knot . The tail of this knot is given by:

 T85(q)=(q;q)2∞∞∑i=0∞∑j=0q(i+i2+j+j2)(q;q)i+j(q;q)2i(q;q)2j. (5.1)

The series is similar to the following -series:

 TΓ=(Ψ(q3,q))2=(q;q)2∞∞∑i=0∞∑j=0q(i+i2+j+j2)(q;q)2i(q;q)2j (5.2)

where is the graph shown on the right handside of Figure 10. This similarity is not surprising since the graph associated to the knot is given in left handside of the Figure 10.

Motivated by this observation, in this section we will study the tail of the family of graphs given in Figure 11 and show the relation between this -series and the false theta function. Note that this graph corresponds to pretzel knots where .

For our tail computations, we will study the element for . Note that when we obtain the bubble element .

Lemma 5.1.

Let , then we have

1. For , we have

where

 En,i1,...,ik=⌈nnnn⌉i1Δ2nΔn+i1k∏j=2⌈nij−1nn⌉ijΔ2nΔn+ij (5.3)
2. For , we have

where

 (5.4)
Proof.

(1) Note first that

We apply the bubble expansion formula times on the previous equation to obtain:

 = = n∑i1=0i1∑i2=0...ik−1∑ik=0⌈nnnn⌉i1Δ2nΔn+i1k∏j=2⌈nij−1nn⌉ij ×

the result then follows.
(2) We apply the bubble expansion formula times and we obtain:

 = = n∑i1=0i1∑i2=0...ik−1∑ik=0⌈nnnn⌉i1Δ2nΔn+i1k−1∏j=2⌈nij−1nn⌉ij ×

Lemma 5.2.
1. Let . Then,

Moreover,

 En,i1,..,ik = (−1)kn+k∑j=1ijqkn/2+k∑j=1(ij(ij/2+1)) × (q;q)4k+2n(q;q)3n−i1+1(q;q)k+12n(q;q)2n+1(q;q)n−i1(q;q)2n−ik(q;q)2ik k∏j=2 (q;q)ij−1−ij+2n+1(q;q)ij−1−ij(q;q)n+ij−1(q;q)2n−ij−1(q;q)n+ij−1+1k∏j=1Δ2nΔn+ij
2. For , we have

Moreover,

 Pn,i1,..,ik = (−1)kn+k∑j=1ijqkn/2+k∑j=1(ij(ij/2+1)) × (q;q)4k+2n(q;q)3n−i1+1(q;q)k+12n(q;q)2n+1(q;q)n−i1(q;q)2n−ik(q;q)2ik k∏j=2 (q;q)ij−1−ij+2n+1(q;q)ij−1−ij(q;q)n+ij−1(q;q)2n−ij−1(q;q)n+ij−1+1k−1∏j=1Δ2nΔn+ij
Proof.
1. Using the fact that

 j∏i=0[n−i]=q(2+3j+j2−2n−2jn)/4(1−q)−1−j(q;q)n(q;q)n−j−1

one obtains :

 ⌈nann⌉b=(−1)n+bqb/2+b2−n/2(q;q)2a(q;q)4n(q;q)1+a−b+2n(q;q)a−b(q;q)2b(q;q)2n(q;q)a+n(q;q)1+a+n(q;q)2−b+n.

This implies,

 = (−1)kn+k∑j=1ijqkn/2+k∑j=1(ij(ij/2+1)) × (q;q)4k+2n(q;q)3n−i1+1(q;q)k+12n(q;q)2n+1(q;q)n−i1(q;q)2n−ik(q;q)2ik k∏j=2 (q;q)ij−1−ij+2n+1(q;q)ij−1−ij(q;q)n+ij−1(q;q)2n−ij−1(q;q)n+ij−1+1

On the other hand, one has

 n∑i1=0i1∑i2=0...ik−1∑ik=0F(i1,...,ik)=n∑ik=0n∑ik−1=ik...n∑i1=i2F(i1,...,ik)

The result then follows.

2. The proof is similar to .

Theorem 5.3.

The tail of the graph is given by

 TΦk,u(q)=(q;q)2∞∞∑l1=0...∞∑lk=0∞∑p1=0...∞∑pu=0g(q;l1,...,lk)g(q;p1,...,pu)(q;q)i+j

where

 g(q;l1,...,lk)=qk∑j=1(ij(ij+1))(q;q)2lkk−1∏j=1(q;q)lj

with .

Proof.

Using Theorem 2.4, we have

 TΦk,u(q)≐nS(n)B(Φk,u)Δn∣∣∣A=q1/4 (5.5)

where is the skein element given in Figure 12.

Using Lemma 5.1, we can write

 (5.6)

Denote the element on the right handside of (5.6) by . Now, Lemma 6.6 in [Hajij1] implies:

 Γn,ik,ju=⌈iknnn−ju⌉0⌈junnn−ik⌉0⌈juiknn⌉0Δik+ju (5.7)

Here,

 ⌈junnn−ik⌉0=(−1)n−ikq(ik−n)/2(q;q)ik+ju(q;q)n(q;q)n+ik(q;q)2n+ju+1(q;q)ik(q;q)2n(q;q)ju+n(q;q)n+ju+ik+1. (5.8)

Moreover Lemma 6.1 in [Hajij1] gives,

A formula for the skein element on the right hand of the previous equation can be found in [MV]. Using this allows us to obtain:

 ⌈