The third term in lens surgery polynomials

Abstract

It is well-known that the second highest coefficient of the Alexander polynomial of any lens space knot in $S^3$ is −1. We show that if the third highest coefficient of the Alexander polynomial $\Delta_K(t)$ of a lens space knot $K$ in $S^3$ is non-zero, then $\Delta_K(t)$ coincides with the Alexander polynomial of the $(2,2g+1)$-torus knot, where $g$ is the Seifert genus of $K$.