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 ${\eta }_{+}$ 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.