#### Volume 20, issue 6 (2016)

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Concordance maps in knot Floer homology

### András Juhász and Marco Marengon

Geometry & Topology 20 (2016) 3623–3673
##### Abstract

We show that a decorated knot concordance $\mathsc{C}$ from $K$ to ${K}^{\prime }$ induces a homomorphism ${F}_{\mathsc{C}}$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\stackrel{̂}{HF}\left({S}^{3}\right)\cong {ℤ}_{2}$ that agrees with ${F}_{\mathsc{C}}$ on the ${E}^{1}$ page and is the identity on the ${E}^{\infty }$ page. It follows that ${F}_{\mathsc{C}}$ is nonvanishing on ${\stackrel{̂}{HFK}}_{0}\left(K,\tau \left(K\right)\right)$. We also obtain an invariant of slice disks in homology 4–balls bounding ${S}^{3}$.

If $\mathsc{C}$ is invertible, then ${F}_{\mathsc{C}}$ is injective, hence

$dim{\stackrel{̂}{HFK}}_{j}\left(K,i\right)\le dim{\stackrel{̂}{HFK}}_{j}\left({K}^{\prime },i\right)$

for every $i,j\in ℤ$. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to ${K}^{\prime }$, then $g\left(K\right)\le g\left({K}^{\prime }\right)$,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 $K$.

##### Keywords
concordance, knot Floer homology, genus
##### Mathematical Subject Classification 2010
Primary: 57M27, 57R58
##### Publication
Revised: 25 January 2016
Accepted: 24 February 2016
Published: 21 December 2016
Proposed: Ciprian Manolescu
Seconded: Peter Teichner, Ronald Stern
##### Authors
 András Juhász Mathematical Institute University of Oxford Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG United Kingdom http://www.maths.ox.ac.uk/people/andras.juhasz Marco Marengon Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ United Kingdom http://www.imperial.ac.uk/people/m.marengon13