Effect of a crossing change on crossing number

TL;DR

This paper investigates how crossing changes affect the crossing number of links, providing estimates under specific conditions and highlighting limitations with examples and counterexamples.

Contribution

It introduces a theorem estimating crossing number changes after crossing modifications, focusing on the span of the X polynomial and the conditions for its applicability.

Findings

Crossing number can be estimated after crossing change under certain conditions.

The theorem applies to alternating links but not non-alternating links.

Counterexamples show the theorem's limitations.

Abstract

The purpose of this article is to give a preliminary clarification on the relation between crossing number and crossing change. With a main focus on the span of X polynomial, we prove that, as our theorem claims, the crossing number of the link after crossing change can be estimated when certain conditions are met. At the end of the article, we give an example to demonstrate a special case for the theorem and a counterexample to explain that the theorem cannot be applied if the obtained link is not alternating.