Applications of the Casson-Walker invariant to the knot complement and the cosmetic crossing conjectures

TL;DR

This paper extends the Casson-Walker invariant formula to 2-component links, providing new examples of knots determined by their complements and contributing to the cosmetic crossing conjecture.

Contribution

It introduces a rational surgery formula for the Casson-Walker invariant of 2-component links, generalizing previous formulas and applying to knot complement uniqueness and the cosmetic crossing conjecture.

Findings

New rational surgery formula for Casson-Walker invariant

Examples of non-hyperbolic L-space where knots are determined by their complements

Applications to the cosmetic crossing conjecture

Abstract

We give a rational surgery formula for the Casson-Walker invariant of a 2-component link in $S^{3}$ which is a generalization of Matveev-Polyak's formula. As application, we give more examples of non-hyperbolic L-space $M$ such that knots in $M$ are determined by their complements. We also apply the result for the cosmetic crossing conjecture.