## Geometry & Topology

### Sutured Floer homology and invariants of Legendrian and transverse knots

#### Abstract

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.

#### Article information

Source
Geom. Topol., Volume 21, Number 3 (2017), 1469-1582.

Dates
Revised: 25 April 2016
Accepted: 17 August 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859206

Digital Object Identifier
doi:10.2140/gt.2017.21.1469

Mathematical Reviews number (MathSciNet)
MR3650078

Zentralblatt MATH identifier
06726508

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R58: Floer homology 57R17: Symplectic and contact topology

#### Citation

Etnyre, John; Vela-Vick, David; Zarev, Rumen. Sutured Floer homology and invariants of Legendrian and transverse knots. Geom. Topol. 21 (2017), no. 3, 1469--1582. doi:10.2140/gt.2017.21.1469. https://projecteuclid.org/euclid.gt/1510859206

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