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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.

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57R58 Floer homology
##### Keywords:
Legendrian knots; transverse knots; Heegaard Floer homology
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