On the equivalence of Legendrian and transverse invariants in knot Floer homology. (English) Zbl 1285.57005

Legendrian and transverse knots have played a prominent role in understanding contact structures on \(3\)-manifolds. There are two classical invariants of Legendrian knots: the Thurston-Bennequin number and the rotation number. Transverse knots possess a single classical invariant called the self-linking number. The first non-classical invariant of Legendrian knots, dubbed Legendrian contact homology (LCH), appeared as an outgrowth of Eliashberg and Hofer’s work on symplectic field theory. In [Geom. Topol. 12, No. 2, 941–980 (2008; Zbl 1144.57012)], P. Ozsváth et al., defined powerful invariants \(\lambda\) and \(\hat\lambda\) of Legendrian links in the standard contact 3-sphere, which take values in the minus and hat versions of knot Floer homology. Their invariants are defined via grid diagrams and are thus combinatorial in nature. Furthermore, \(\lambda\) and \(\hat\lambda\) remain unchanged under negative Legendrian stabilization, and, therefore, give rise to transverse link invariants \(\theta\) and \(\hat\theta\) through Legendrian approximation. \(\lambda\), \(\hat\lambda\), \(\theta\), \(\hat\theta\) are referred to as the GRID invariants. P. Lisca et al. [J. Eur. Math. Soc. (JEMS) 11, No. 6, 1307–1363 (2009; Zbl 1232.57017)] used open book decompositions to define invariants \({\mathcal L}\) and \(\hat{\mathcal L}\) of Legendrian knots in arbitrary contact \(3\)-manifolds. These alternate invariants also take values in the minus and hat version of knot Floer homology. Both \({\mathcal L}\) and \(\hat{\mathcal L}\) remain unchanged under negative Legendrian stabilization, and may therefore be used as above to define transverse knot invariants \({\mathcal T}\) and \(\hat{\mathcal T}\) via Legendrian approximation. All four \({\mathcal L}\), \(\hat{\mathcal L}\), \({\mathcal T}\), \(\hat{\mathcal T}\) are referred to as the LOSS invariants. It has been conjectured for several years that the GRID invariants agree with the LOSS invariants where both are defined – for Legendrian and transverse knots in the tight contact \(3\)-sphere \((S^3,\xi_{\text{std}})\).
In this paper, the authors define a third set of invariants of transverse links in arbitrary contact \(3\)-manifolds. They are denoted by \(t\) and \(\hat t\) and referred to as BRAID invariants. The authors show that these third invariants agree with the two mentioned above. They prove that if \(K\) is a transverse knot in the standard contact \(3\)-sphere, then there exists an isomorphism of bigraded \((\mathbb{Z}/2\mathbb{Z})[U]\)-modules, \(\psi:HFK^-(-S^3,K)\to HFK^-(-S^3,K)\) which sends \({\mathcal T}(K)\) to \(\theta(K)\). They prove it by showing that LOSS=BRAID and BRAID=GRID.


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
Full Text: DOI arXiv


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