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Sutured Floer homology and invariants of Legendrian and transverse knots

John B Etnyre, David Shea Vela-Vick and Rumen Zarev

Geometry & Topology 21 (2017) 1469–1582

Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus versions of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology. In addition, to a Legendrian or transverse knot K (Y,ξ) we associate distinguished classes EH(K) HFK(Y,K) and EH(K) HFK+(Y,K), which are each invariant under Legendrian or transverse isotopies of K. The distinguished class EH is shown to agree with the Legendrian/transverse invariant defined by Lisca, Ozsváth, Stipsicz and Szabó despite a strikingly dissimilar definition. While our definitions and constructions only involve sutured Floer homology and contact geometry, the identification of our invariants with known invariants uses bordered sutured Floer homology to make explicit computations of maps between sutured Floer homology groups.

Legendrian knots, transverse knots, Heegaard Floer homology
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57R58, 57R17
Received: 4 September 2014
Revised: 25 April 2016
Accepted: 17 August 2016
Published: 10 May 2017
Proposed: Yasha Eliashberg
Seconded: András I. Stipsicz, Peter S. Ozsváth
John B Etnyre
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
United States
David Shea Vela-Vick
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803
United States
Rumen Zarev
Department of Mathematics
University of California, Berkeley
970 Evans Hall
Berkeley, CA 94720-3840
United States