Invariants of genus 2 mutants

TL;DR

This paper explores the invariants of genus 2 mutant knots, providing examples that distinguish them using various polynomial invariants and Vassiliev invariants, revealing new differences not previously observed.

Contribution

It presents the first known examples of genus 2 mutants with different Kauffman polynomials and Homfly polynomials under specific conditions, advancing understanding of their invariants.

Findings

Examples of genus 2 mutants with different Kauffman polynomials.

Examples with differing Homfly polynomials when v=s^3.

Differences in Vassiliev invariants of degree 7.

Abstract

Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 3-variable polynomials, answering a question raised by Dunfield et al in their study of genus 2 mutants. While pairs of genus 2 mutant knots have the same Jones polynomial, given from the Homfly polynomial by setting v=s^2, we give examples whose Homfly polynomials differ when v=s^3. We also give examples which differ in a Vassiliev invariant of degree 7, in contrast to satellites of Conway mutant knots.