Concordance maps in knot Floer homology

TL;DR

This paper demonstrates how decorated knot concordances induce homomorphisms on knot Floer homology that preserve gradings and spectral sequence structures, leading to new invariants and inequalities related to knot genus and fibredness.

Contribution

It introduces a new homomorphism induced by knot concordance on knot Floer homology, preserving gradings and spectral sequence structures, and derives implications for knot genus and fibredness.

Findings

Homomorphism $F_C$ preserves Alexander and Maslov gradings.

$F_C$ induces a spectral sequence morphism compatible with $ ext{HF}(S^3)$.

Invertible concordances imply inequalities on knot genus and fibredness.

Abstract

We show that a decorated knot concordance $C$ from $K$ to $K'$ induces a homomorphism $F_C$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}(S^3) \cong \mathbb{Z}_2$ that agrees with $F_C$ on the $E^1$ page and is the identity on the $E^\infty$ page. It follows that $F_C$ is non-vanishing on $\widehat{HFK}_0(K, \tau(K))$. We also obtain an invariant of slice disks in homology 4-balls bounding $S^3$. If $C$ is invertible, then $F_C$ is injective, hence $\dim \widehat{HFK}_j(K,i) \le \dim \widehat{HFK}_j(K',i)$ for every $i$, $j \in \mathbb{Z}$. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to $K'$, then $g(K) \le g(K')$, where $g$ denotes the Seifert genus. Furthermore, if $g(K) = g(K')$ and $K'$ is fibred, then so is $K$.