# On the Cosmetic Crossing Conjecture for Special Alternating Links

## TL;DR

This paper proves that certain special alternating links, including all special alternating knots, cannot be altered by non-trivial crossing changes without changing their isotopy type, advancing understanding of the cosmetic crossing conjecture.

## Contribution

It extends the class of links for which the cosmetic crossing conjecture is confirmed, combining techniques from L-space knot theory and classical crossing change analysis.

## Key findings

- Special alternating links do not admit non-nugatory crossing changes preserving isotopy.
- The proof uses results on L-space branched double-covers and classical techniques from the unknot.
- The work confirms the cosmetic crossing conjecture for a broad family of links.

## Abstract

We prove that a family of links, which includes all special alternating knots, does not admit non-nugatory crossing changes which preserve the isotopy type of the link. Our proof incorporates a result of Lidman and Moore on crossing changes to knots with $L$-space branched double-covers, as well as tools from Scharlemann and Thompon's proof of the cosmetic crossing conjecture for the unknot.