A note on changemaker lattices and Alexander polynomials of lens space knots

TL;DR

This paper provides an alternative proof of a theorem relating lens space surgeries on knots to Alexander polynomials, utilizing changemaker lattices to establish new constraints on these polynomials.

Contribution

It introduces a novel proof technique using changemaker lattices to analyze Alexander polynomials of knots with lens space surgeries, extending previous results.

Findings

Constraints on Alexander polynomials when certain coefficients are non-zero

New proof of Tange's theorem using lattice technology

Insights into the structure of lens space knots

Abstract

We give an alternative proof of a recent theorem of Tange using the technology of changemaker lattices. Specifically, for $K\subset S^3$ a non-trivial knot with a lens space surgery, we give constraints on the Alexander polynomial of $K$ and the lens space surgery when the coefficient of $T^{g(K)-2}$ in $\Delta_K(T)$ is non-zero.