Behavior of knot invariants under genus 2 mutation

Abstract

Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares genus 2 mutation with the better-known Conway mutation in the context of knots in the 3-sphere. Despite the fact that any Conway mutation can be achieved by a sequence of at most two genus 2 mutations, the invariants that are preserved by genus 2 mutation are a proper subset of those preserved by Conway mutation. In particular, while the Alexander and Jones polynomials are preserved by genus 2 mutation, the HOMFLY-PT polynomial is not. In the case of the sl_2-Khovanov homology, which may or may not be invariant under Conway mutation, we give an example where genus 2 mutation changes this homology. Finally, using these techniques, we exhibit examples of knots with the same same colored Jones polynomials, HOMFLY-PT polynomial, Kauffman polynomial, signature and volume, but different Khovanov homology.