When will the crossing number of an alternating link decrease by two via a crossing change?

TL;DR

This paper characterizes when a crossing change in a reduced alternating link diagram decreases the crossing number by exactly two, using Tutte polynomial analysis of associated plane graphs.

Contribution

It provides a simple necessary and sufficient condition for the crossing number decrease, linking link diagram changes to Tutte polynomial properties of plane graphs.

Findings

Identifies when crossing number decreases by two after a crossing change.

Establishes a condition based on Tutte polynomial analysis.

Connects link diagram modifications to graph polynomial behavior.

Abstract

Let $D$ be a reduced alternating diagram of a non-split link $L$ and $\tilde{L}$ be the link whose diagram is obtained from $D$ by a crossing change. If $\tilde{L}$ is alternating, then $c(\tilde{L})\leq c(L)-2$. In this paper we explore when $c(\tilde{L})=c(L)-2$ holds and obtain a simple sufficient and necessary condition in terms of plane graphs corresponding to $L$. This result is obtained via analyzing the behavior of the Tutte polynomial of the signed plane graph corresponding to $\tilde{L}$.