The $A$-polynomial of the $(-2,3,3+2n)$ pretzel knots

Abstract

We show that the A-polynomial $A_n$ of the 1-parameter family of pretzel knots $K_n=(-2,3,3+2n)$ satisfies a linear recursion relation of order 4 with explicit constant coefficients and initial conditions. Our proof combines results of Tamura-Yokota and the second author. As a corollary, we show that the $A$-polynomial of $K_n$ and the mirror of $K_{-n}$ are related by an explicit $\GL(2,\BZ)$ action. We leave open the question of whether or not this action lifts to the quantum level.