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Sutured Floer homology and invariants of Legendrian and transverse knots. (English) Zbl 1420.57035
Summary: 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 \(K \subset (Y,\xi)\) we associate distinguished classes \(\underrightarrow{\operatorname{EH}}(K) \in \operatorname{HFK}^-(-Y,K)\) and \(\underleftarrow{\operatorname{EH}}(K) \in \operatorname{HFK}^+(-Y,K)\), which are each invariant under Legendrian or transverse isotopies of \(K\). The distinguished class \(\underrightarrow{\operatorname{EH}}\) is shown to agree with the Legendrian/transverse invariant defined by Lisca, Ozsváth, Stipsicz and Szabó [P. Lisca et al., J. Eur. Math. Soc. (JEMS) 11, No. 6, 1307–1363 (2009; Zbl 1232.57017)] 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.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
57R17 Symplectic and contact topology in high or arbitrary dimension
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