The Non-Commutative a-Polynomial of Twist Knots

Abstract

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots.Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form J(n) = P k c(n, k)Ĵ(k) given a recursion relation for (Ĵ (n)) and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with −15 and 15 crossings.The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

Super-A-polynomials for twist knots

“…Classical A-polynomials for the twist knots. The results were first obtained in [66] where x = M 2 and y = L. (See also Appendix C in [67].) These are equal to the classical super-Apolynomials at a = 1 and t = −1.…”

“…1. Furthermore, we have varified that, for these knots, the quantum super-A-polynomials A super (x,ŷ, q, a, t) reduce to the quantum A-polynomials A(x,ŷ; q) obtained in [67].…”

“…However, as pointed out in [67], the package qMultisum.m does not give a difference equation of minimal order in general.…”

“…7 The algorithm which computes difference equations of minimal order for multi-sum expressions has been developed by S. Garoufalidis and X. Sun [67,72,73]. By the algorithm with a certicate, the quantum A-polynomials were obtained for the twist knots A(K p ;x,ŷ; q) in the range −14 ≤ p ≤ 15 [67].…”

“…Sun [67,72,73]. By the algorithm with a certicate, the quantum A-polynomials were obtained for the twist knots A(K p ;x,ŷ; q) in the range −14 ≤ p ≤ 15 [67]. Similarly to the generalized volume conjecture in §3.1, the AJ conjecture was generalized to the following form [26] for colored superpolynomials.…”