Genus two mutant knots with the same dimension in knot Floer and Khovanov homologies

TL;DR

This paper constructs an infinite family of knots that share the same total homology dimensions in knot Floer and Khovanov theories, yet are distinguished by their bigraded structures, revealing nuanced mutation effects.

Contribution

It introduces an infinite family of genus two mutant knots with identical total homology dimensions but distinct bigraded homology groups, highlighting subtle mutation invariants.

Findings

Knots with identical total homology dimensions but different bigraded structures.

Genus two mutation swaps homology groups in specific gradings.

Mutant knots are distinguishable by their bigraded homology groups.

Abstract

We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $\delta$-gradings $k$ and $-k$.