Volume 20, issue 6 (2016)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 C from K to K induces a homomorphism FC on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to HF̂(S3)2 that agrees with FC on the E1 page and is the identity on the E page. It follows that FC is nonvanishing on HFK̂0(K,τ(K)). We also obtain an invariant of slice disks in homology 4–balls bounding S3.

If C is invertible, then FC is injective, hence

dimHFK̂j(K,i) dimHFK̂j(K,i)

for every i,j . This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot K to K, then g(K) g(K), where g denotes the Seifert genus.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