On the AJ conjecture for cables of the figure eight knot

TL;DR

This paper proves the AJ conjecture for (r,2)-cables of the figure eight knot when r is an odd integer with |r| ≥ 9, extending the class of knots for which the conjecture holds.

Contribution

It establishes the AJ conjecture for a new family of cabled knots, specifically (r,2)-cables of the figure eight knot with certain odd r values.

Findings

AJ conjecture verified for (r,2)-cables with |r| ≥ 9

Extends known cases of the AJ conjecture to new cabled knots

Supports the conjecture's validity for broader classes of knots

Abstract

The AJ conjecture relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been verified for some classes of knots, including all torus knots, most double twist knots, (-2,3,6n \pm 1)-pretzel knots, and most cabled knots over torus knots. In this paper we study the AJ conjecture for (r,2)-cables of a knot, where r is an odd integer. In particular, we show that the AJ conjecture holds true for (r,2)-cables of the figure eight knot, where r is an odd integer satisfying |r| \ge 9.