Twisting, mutation and knot Floer homology

TL;DR

This paper proves the stabilization of knot Floer homology for knots with increasing positive twists and constructs infinite families of prime, positive mutant knots with identical knot Floer homology groups, revealing deep invariance properties.

Contribution

It demonstrates the stabilization of knot Floer homology under positive twisting and constructs infinite families of prime, positive mutant knots with isomorphic Floer homology groups.

Findings

Knot Floer homology stabilizes as twists increase.

Constructs infinite families of prime, positive mutant knots with identical Floer homology.

Describes how to generate positive mutants with matching invariants.

Abstract

Let $\mathcal{L}$ be a knot with a fixed positive crossing and $\mathcal{L}_n$ the link obtained by replacing this crossing with $n$ positive twists. We prove that the knot Floer homology $\widehat{\text{HFK}}(\mathcal{L}_n)$ `stabilizes' as $n$ goes to infinity. This categorifies a similar stabilization phenomenon of the Alexander polynomial. As an application, we construct an infinite family of prime, positive mutant knots with isomorphic bigraded knot Floer homology groups. Moreover, given any pair of positive mutants, we describe how to derive a corresponding infinite family positive mutants with isomorphic bigraded $\widehat{\text{HFK}}$ groups, Seifert genera, and concordance invariant $\tau$.