Concordance maps in knot Floer homology

Juhász, András; Marengon, Marco

doi:10.2140/gt.2016.20.3623

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Concordance maps in knot Floer homology

András Juhász and Marco Marengon

Geometry & Topology 20 (2016) 3623–3673

DOI: 10.2140/gt.2016.20.3623

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 $\hat{HF} (S^{3}) ≅ ℤ_{2}$ 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 nonvanishing on ${\hat{HFK}}_{0} (K, τ (K))$ . 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 {\hat{HFK}}_{j} (K, i) \leq dim {\hat{HFK}}_{j} (K^{'}, i)

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 This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to $K^{'}$ , then $g (K) \leq g (K^{'})$ , where $g$ denotes the Seifert genus.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

Keywords

concordance, knot Floer homology, genus

Mathematical Subject Classification 2010

Primary: 57M27, 57R58

References

Publication

Received: 18 September 2015

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

Volume 20, issue 6 (2016)