The non-commutative $A$-polynomial of twist knots

TL;DR

This paper introduces a multivariable creative telescoping method and applies it to compute the non-commutative $A$-polynomial of specific twist knots, advancing the understanding of quantum invariants and knot topology.

Contribution

The paper develops a new multivariable creative telescoping technique and explicitly computes the non-commutative $A$-polynomial for twist knots with -8 and 11 crossings.

Findings

Successfully computed non-commutative $A$-polynomials for specific twist knots.

Demonstrated the effectiveness of the telescoping method in quantum topology.

Connected the non-commutative $A$-polynomial to classical knot invariants.

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 $J(n)=\sum_k c(n,k) \hatJ (k)$ given a recursion relation for $(\hatJ(n))$ a the hypergeometric kernel $c(n,k)$. As an application of our method, we explicitly compute the non-commutative $A$-polynomial for twist knots with -8 and 11 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.