The AJ-Conjecture for Cables of Two Bridge Knots

TL;DR

This paper investigates conditions under which the AJ-conjecture, relating the A-polynomial and colored Jones polynomial, holds for cables of two-bridge knots, extending known results to a broader class of knots.

Contribution

It provides sufficient conditions for cable knots of two-bridge knots to satisfy the AJ-conjecture, linking diagram crossings to conjecture validity.

Findings

AJ-conjecture holds for certain cable knots under specified conditions.

The paper establishes a relation between crossing numbers and the conjecture's validity.

Conditions depend on the signs and counts of crossings in the knot diagram.

Abstract

The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If a two-bridge knot $K$ satisfies the $AJ$-conjecture, we give sufficient conditions on $K$ for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has $\eta_+$ positive crossings and $\eta_-$ negative crossings, then $C$ will satisfy the $AJ$-conjecture when $(r+4\eta_-)(r-4\eta_+)>0$ and the conditions of the main theorem are satisfied.