The AJ-Conjecture for Cables of Two Bridge Knots

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 $η_+$ positive crossings and $η_-$ negative crossings, then $C$ will satisfy the $AJ$-conjecture when $(r+4η_-)(r-4η_+)>0$ and the conditions of the main theorem are satisfied.

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