Behavior of knot invariants under genus 2 mutation
The process of genus two mutation involves cutting a 3-manifold along an embedded closed genus 2 surface \(F\), twisting by a specific type of involution, \( \tau \), and gluing back the surface. These involutions have the property that if \( \gamma \) is a non-boundary parallel simple closed curve then \( \gamma ( \tau)\) is isotopic to \( \gamma \). The topological type of \(F\) is denoted by \((g,s)\) where \(g\) is the genus and \(s\) denotes the number of boundary components. The authors compare genus two mutation and Conway mutation on knots in \(S^3\). Although any Conway mutation can be expressed as the composition of two genus two mutations, the invariants preserved by genus two mutation are a subset of those preserved by Conway mutation. The authors provide substantial background on the subject of mutation. The authors give a survey of earlier results on the topology of knot mutation. They note that the Alexander polynomial and generalized signatures are preserved by (2,0) mutation and examine the behavior of quantum invariants under (2,0) mutation. The authors show that the colored Jones polynomials are invariant under (2,0) mutation for all colors, generalizing the approach of Morton and Tracyzk. The authors give an example of a pair of knots with different HOMFLY-PT polynomials that are related by (2,0) mutation, demonstrating the non-invariance of the HOMFLY-PT polynomial. They prove that \(sl_2 \) Khovanov homology is not invariant under genus two mutation by exhibiting pairs of knots related by genus 2 mutation with different homologies. Finally, the authors prove that there exist knots with the same colored Jones polynomials (for all colors), HOMFLY-PT and Kauffman polynomials, volume and signature but different Khovanov homologies.
scientific article; zbMATH DE number 5759886