On the AJ conjecture for cables of twist knots

TL;DR

This paper verifies the AJ conjecture, linking the A-polynomial and colored Jones polynomial, for specific cable knots derived from twist knots under certain conditions.

Contribution

It confirms the AJ conjecture for (r,2)-cables of m-twist knots for all odd r satisfying particular inequalities.

Findings

AJ conjecture holds for specified cable knots.

Conditions on r depend on the twist parameter m.

Results extend understanding of the AJ conjecture in knot theory.

Abstract

We study the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot in $S^3$. We confirm the AJ conjecture for $(r,2)$-cables of the $m$-twist knot, for all odd integers $r$ satisfying $\begin{cases} (r+8)(r-8m)>0 &{if~} m> 0, \\ r(r+8m-4)>0 &{if~} m<0.\end{cases} $