The non-commutative A-polynomial of (-2,3,n) pretzel knots

TL;DR

This paper explicitly computes the non-commutative A-polynomial for a family of pretzel knots using a guessing method, supporting the AJ Conjecture and enabling efficient Kashaev invariant calculations.

Contribution

It provides explicit non-commutative A-polynomials for (-2,3,3+2p) pretzel knots, supporting conjectures and enabling rapid invariant computations.

Findings

Computed non-commutative A-polynomials for specific pretzel knots.

Supports the AJ Conjecture for knots with reducible A-polynomial.

Enables linear-time numerical computation of the Kashaev invariant.

Abstract

We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the "method of guessing", we obtain this polynomial explicitly for the K_p = (-2, 3, 3+2p) pretzel knots for p = -5, ..., 5. This is a particularly interesting family since the pairs (K_p, -K_{-p}) are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the non-commutative A-polynomial (a) complements the computation of the A-polynomial of the pretzel knots done by the first author and Mattman, (b) supports the AJ Conjecture for knots with reducible A-polynomial and (c) numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariant to numerically verify the Volume Conjecture for the above mentioned pretzel knots.