Irreducibility of q-difference operators and the knot 7_4

TL;DR

This paper computes the minimal recurrence for the colored Jones polynomial of the 7_4 knot, verifies the AJ Conjecture for it, and introduces new algebraic techniques for analyzing q-difference operators.

Contribution

It introduces an improved irreducibility criterion for q-difference operators and applies symbolic summation to knot invariants, verifying the AJ Conjecture for the 7_4 knot.

Findings

Minimal-order recurrence for 7_4 knot's colored Jones polynomial computed

AJ Conjecture verified for the 7_4 knot

New irreducibility criterion for q-difference operators developed

Abstract

Our goal is to compute the minimal-order recurrence of the colored Jones polynomial of the 7_4 knot, as well as for the first four double twist knots. As a corollary, we verify the AJ Conjecture for the simplest knot 7_4 with reducible non-abelian SL(2,C) character variety. To achieve our goal, we use symbolic summation techniques of Zeilberger's holonomic systems approach and an irreducibility criterion for q-difference operators. For the latter we use an improved version of the qHyper algorithm of Abramov-Paule-Petkovsek to show that a given q-difference operator has no linear right factors. En route, we introduce exterior power Adams operations on the ring of bivariate polynomials and on the corresponding affine curves.