Alternating knots do not admit cosmetic crossings

TL;DR

This paper proves that alternating knots cannot have cosmetic crossings by analyzing homology groups of related 4-manifolds and applying results from Heegaard Floer homology, confirming the cosmetic crossing conjecture for certain classes of knots.

Contribution

It introduces a homological approach to prove the non-existence of cosmetic crossings in alternating knots, extending previous results using Heegaard Floer theory.

Findings

Alternating knots do not admit cosmetic crossings.

Homology groups of associated 4-manifolds reveal co-primality conditions.

Supports the cosmetic crossing conjecture for knots with Heegaard Floer L-space double covers.

Abstract

By examining the homology groups of a 4-manifold associated to an integral surgery on a knot $K$ in a rational homology 3-sphere $Y$ yielding a rational homology 3-sphere $Y^*$ with surgery dual knot $K^*$, we show that the subgroups generated by $[K]$ and $[K^*]$ in $H_1(Y)$ and $H_1(Y^*)$, respectively, have co-prime orders. We obtain an immediate corollary that, in conjunction with an argument of Lidman and Moore, proves the cosmetic crossing conjecture for knots whose branched double covers are Heegaard Floer L-spaces.