Concordance maps in knot Floer homology

Abstract

We show that a decorated knot concordance $C$ from $K$ to $K^{\prime }$ 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 $\dim {\widehat{HFK}}_j(K,i)\le \dim {\widehat{HFK}}_j(K^{\prime },i)$ 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 This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to $K^{\prime }$, then $g(K)\le g(K^{\prime })$, where $g$ denotes the Seifert genus. Furthermore, if $g(K)=g(K^{\prime })$ and $K^{\prime }$ is fibred, then so is $K$.

38 pages, 3 figures, to appear in Geometry and Topology

References in corpus (1)

• Knot Floer homology detects fibred knots

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