N. Dunfield, S. Garoufalidis, A. Shumakovitch and M. Thislethwaite
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 two genus 2 mutations, the invariants that are preserved by genus 2 mutation is a proper subset of those preserved by Conway mutation. In particular, we show that while the Alexander and Jones polynomials are preserved by genus 2 mutation, the HOMFLY-PT and (perhaps) the Kauffman polynomials are 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.
Key words: mutation, symmetric surfaces, Khovanov Homology, volume, colored Jones polynomial, HOMFLY polynomial, Kauffman polynomial, signature.
Notes: 16 pages, 12 figures.