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

TL;DR

This paper characterizes certain alternating knots by their Alexander polynomial coefficients, proving that specific coefficient conditions imply the knot is a particular torus knot, thus supporting the Fox Trapezoidal Conjecture.

Contribution

It establishes a new characterization of $T_{2g+1, ext{±}2}$ among alternating knots based on Alexander polynomial coefficients, advancing understanding of knot invariants.

Findings

If $|a_g|=|a_{g-1}|$, then $K$ is the torus knot $T_{2g+1, ext{±}2}$.

Supports the Fox Trapezoidal Conjecture for this class of knots.

Uses Ozsváth and Szabó's work on alternating knots to prove the result.

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.