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Concordance maps in knot Floer homology

András Juhász and Marco Marengon

Geometry & Topology 20 (2016) 3623–3673

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 This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot K to K, then g(K) g(K), where g denotes the Seifert genus.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

concordance, knot Floer homology, genus
Mathematical Subject Classification 2010
Primary: 57M27, 57R58
Received: 18 September 2015
Revised: 25 January 2016
Accepted: 24 February 2016
Published: 21 December 2016
Proposed: Ciprian Manolescu
Seconded: Peter Teichner, Ronald Stern
András Juhász
Mathematical Institute
University of Oxford
Andrew Wiles Building, Radcliffe Observatory Quarter
Woodstock Road
United Kingdom
Marco Marengon
Department of Mathematics
Imperial College London
180 Queen’s Gate
United Kingdom