Concordance maps in knot Floer homology

Abstract

We show that a decorated knot concordance $\mathcal{C}$ from $K$ to $K^{\prime }$ induces a homomorphism $F_{\mathcal{C}}$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}\left(S^3\right)\cong {\mathbb{Z}}_2$ that agrees with $F_{\mathcal{C}}$ on the $E^1$ page and is the identity on the $E^{\infty }$ page. It follows that $F_{\mathcal{C}}$ is nonvanishing on ${\widehat{HFK}}_0\left(K,\tau \left(K\right)\right)$. We also obtain an invariant of slice disks in homology 4–balls bounding $S^3$.

If $\mathcal{C}$ is invertible, then $F_{\mathcal{C}}$ is injective, hence

$$dim{\widehat{HFK}}_j\left(K,i\right)\le dim{\widehat{HFK}}_j\left(K^{\prime },i\right)$$

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 This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to $K^{\prime }$, then $g\left(K\right)\le g\left(K^{\prime }\right)$, where $g$ denotes the Seifert genus. Furthermore, if $g\left(K\right)=g\left(K^{\prime }\right)$ and $K^{\prime }$ is fibred, then so is $K$.