Torus knots and quantum modular forms

In this paper we compute a $q$-hypergeometric expression for the cyclotomic expansion of the colored Jones polynomial for the left-handed torus knot $(2,2t+1)$ and use this to define a family of quantum modular forms which are dual to the generalized Kontsevich-Zagier series.


Introduction and Statement of Results
Several years ago Zagier introduced quantum modular forms [30]. These are functions defined for z ∈ Q (or equivalently, for q a root of unity, where q := e 2πiz ) which behave like modular forms. For g(z) to be quantum modular means that instead of requiring that g(z) − χ(γ) (cz + d) −k g az+b cz+d = 0 for γ = ( a b c d ) ∈ SL 2 (Z) as with classical modular forms, we only ask that g(z) − χ(γ) (cz + d) −k g az+b cz+d have "nice" properties such as continuity or analyticity. The word quantum refers to the fact that these functions have "the 'feel' of the objects in perturbative quantum field theory" [30, p. 659]. A celebrated example is the Kontsevich-Zagier series [29] F (q) := ∞ n=0 (q) n , (1.1) where we use the standard notation, (a) n := (1 − a)(1 − aq) · · · (1 − aq n−1 ). (1.2) Note that F (q) does not converge on any open subset of C, but it is well-defined at any roots of unity.
Based on the colored Jones polynomial for the torus knot T (2,2t+1) at roots of unity, the first author [14] introduced a family of quantum modular forms generalizing F (q) (1.1), (1.8) Here n k q is the usual q-binomial coefficient, Note that when t = 1 we recover the Kontsevich-Zagier series, F 1 (q) = q F (q). Our purpose in this article is to use the perspective of quantum invariants to generalize U (x; q) and (1.4). As a dual to F t (q), we make the following definition.
(1.10) This function is well-defined for |q| < 1. When x = −1 it is also defined when q is a root of unity. Our first main result is the following generalization of (1.4).
Our second main result is a Hecke-type expansion for U t (x; q). (The case t = 1 and x = −1 appears in [9].) (1.13) The paper is constructed as follows. In Section 2, we review Bailey pairs and their relation to the colored Jones polynomial. In Section 3 we study the colored Jones polynomial for the torus knot T (2,2t+1) . In particular, we use the Bailey pair machinery to compute the coefficients of the cyclotomic expansion of J N (T * (2,2t+1) ; q), which leads to Theorem 1.2. In Section 4 we prove Theorem 1.3, again using the Bailey machinery. In Section 5 we extend Theorems 1.2 and 1.3 to the vector-valued setting. We close with some suggestions for future research and an appendix containing some examples.

Bailey Pairs and The Colored Jones Polynomial
In this section we review facts about Bailey pairs and their relation to the colored Jones polynomial.
First recall [2] that two sequences (α n , β n ) form a Bailey pair relative to a if or equivalently, The Bailey lemma [2] states that if (α n , β n ) is a Bailey pair relative to a, then so is (α n , β n ), where In particular, if b, c → ∞ then we have α n = a n q n 2 α n (2.5) and Next recall the cyclotomic expansion of the colored Jones polynomial due to Habiro [13] where we have The colored Jones polynomial J N (K; q) and the coefficients C n (K; q) defined in (2.8) can be regarded as a Bailey pair (α n , β n ) relative to q 2 . Namely, comparing equations (2.8) and (2.2) we have (see also [13,16]) β n = q −n C n (K; q).

The Colored Jones Polynomial for Torus Knots
For some knots K, explicit forms of J N (K; q) and/or C N (K; q) are known in the literature. For instance, when K is the right-handed torus knot T (s,t) , where s and t are coprime positive integers, the colored Jones polynomial is given by [25,27] J N (T (s,t) ; q) = q Using difference equations, the first author [15] constructed a q-hypergeometric expression for J N (T (s,t) ; q) when s = 2, (See [18] for similar expressions for some other torus knots.) Comparing this with the generalized Kontsevich-Zagier series (1.8), we find that J N (T (2,2t+1) ; q) and F t (q) agree at roots of unity, With (1.5) and (2.8) in mind, we see that to discover Definition 1.1 and prove Theorem 1.2 we need to compute the cyclotomic expansion of the colored Jones polynomial of the left-handed torus knot T * (2,2t+1) . Recalling that the colored Jones polynomial for the mirror image K * is given from that for K (1.5), we find from (3.1) that Then the coefficients C n in the cyclotomic expansion (2.8) are given from the inverse transform (2.11) as In the following proposition we give a q-hypergeometric expression for the coefficients C n (T * (2,2t+1) ; q). We use the usual characteristic function χ(X) := 1, when X is true, 0, when X is false.
While the above proposition does furnish an attractive q-hypergeometric expression for C n (T * (2,2t+1) ; q), it is not apparent that these coefficients are Laurent polynomials in q, as guaranteed by (2.9). This is made clear with the next proposition.
Proof. We recall the classical q-binomial identity, (3.17) Using this identity, we also have, for arbitrary c, We may use the two identities (3.17) and (3.18) to transform (3.6) into (3.14) as follows. First, if n t+1 = n t−1 then the sum in (3.6) vanishes, so we may assume n t+1 > n t−1 . The sum over n t is n t+1 and the second part of identity (3.18) then enables us to evaluate this sum, giving We then set n t+1 = n t−1 + k 1 , with k 1 ≥ 1. The sum over n t−1 is and (3.17) allows us to evaluate this sum, resulting in We continue in the same manner, next setting n t+2 = n t−2 + k 2 , with k 2 ≥ k 1 , and we may then take the sum over n t−2 using (3.17). Iterating this process (taking the sum over n t−a after setting n t+a = n t−a + k a ), we arrive at (3.14). The expression in equation (3.15) follows from the fact that We are now prepared to prove the duality in Theorem 1.2.

Hecke-Type Formulae
In this section we prove Theorem 1.3 as well as a simpler formula when t = 1.

(4.2)
This gives Using (1 − x 2r+1 ) to split the right-hand side into two sums and then replacing (r, n) by (−r − 1, −n − 1) in the second sum yields the result.
This is a limiting value of the Eichler integral of a vector-valued modular form Φ where as usual q = e 2πiτ , and (a, b, · · · ; q) ∞ = (a) ∞ (b) ∞ · · · . Note that these q-series appeared in the Andrews-Gordon identities [1], the t = 2 case of which corresponds to the Rogers-Ramanujan identities. The quantum modularity of F In this section we define q-series U (−1; ζ N ) (see Theorem 5.5) and we find Hecke-type formulae for U (m) t (q) (see Theorem 5.6). We begin by defining an analogue of the colored Jones polynomial, When m = 1 this coincides with the colored Jones polynomial J (t,1) 2t+1) ; q).
Next we study a cyclotomic expansion for J (t,m) N (q), Equation (2.11) shows that we have We note that C (t,1) n (q) = C n (T * (2,2t+1) ; q). The next two propositions are generalizations of Propositions 3.1 and 3.2, respectively.

(5.11)
Proof. In light of equation (5.10) and the definition of a Bailey pair (2.1), we need to find β n such that The proof of this Bailey pair is exactly as in the proof of Proposition 3.1 but with = t − m instead of t − 1.
Proposition 5.3. We have (5.14) Proof. The proof is similar to that of Proposition 3.2. We begin by treating the sum over n t , which is n t+1 Assuming for the moment that n t+1 > n t−1 , the second part of identity (3.18) enables us to evaluate this sum, giving The rest of the proof is the same as the proof of Proposition 3.2. The only difference between equations (3.19) and (5.16) is that the latter contains the term q − t−m i=1 n i instead of the term q − t−1 i=1 n i , which results in (5.13) instead of (3.14).
Now suppose that n t+1 = n t−1 . The sum (5.15) on n t is trivial and reduces to (−1) n t−1 q −( n t−1 +1 2 ) (1 − q n t−1 −χ(t>m)n t−m ). (5.17) This corresponds to k 1 = 0, and the sum on n t−1 is then If n t−2 = n t+2 then we collapse the sum again and obtain k 2 = 0, continuing in this way until n t+a > n t−a , and then applying (3.17) and arguing as usual. Note that if n t+m = n t−m then the sum vanishes, so we have k m ≥ 1.
Using the expression for C (t,m) n (q), we are now prepared to generalize Definition 1.1.
Proof. We have J With the help of (5.7), we get (5.20).