A characterization of $T_{2g+1,2}$ among alternating knots

Abstract

Let $K$ be a genus $g$ alternating knot with Alexander polynomial $\Delta_K(T)=\sum_{i=-g}^ga_iT^i$. We show that if $|a_g|=|a_{g-1}|$, then $K$ is the torus knot $T_{2g+1,\pm2}$. This is a special case of the Fox Trapezoidal Conjecture. The proof uses Ozsv\'ath and Szab\'o's work on alternating knots.