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Augmentations and rulings of Legendrian knots. (English) Zbl 1082.57020

The work studies relations between the two types of invariants of Legendrian knots \(K\) in the standard contact \(\mathbb R^3\). The first type is the contact homology of the Differential Graded Algebras of \(K\) (Chekanov-Eliashberg DGA). The second type is the count of rulings (special decompositions of the front projection of \(K\)) introduced in the works of Chekanov, Fuchs, and Pushkar. The DGA is hard to compute, its linearizations that come from augmentations are more handy. Here an augmentation is a map from the DGA to the base ring that sends the image of the differential to zero. Let \(r(K)\) be the rotation number of a Legendrian knot \(K\) and let \(\rho\) be a divisor of \(2r(K).\) If an augmentation has support on generators of grading zero \(\mod 2r(K)\) (resp. zero \(\mod \rho\)), then it is called a graded (resp, \(\rho\)-graded) augmentation. If a front diagram of \(K\) has a graded (resp. \(\rho\)-graded) ruling, then the DGA of \(K\) has a graded (resp. \(\rho\)-graded augmentation) [see D. Fuchs, J. Geom. Phys. 47, No. 1, 43–65 (2003; Zbl 1028.57005)].
The author proves that if the DGA of \(K\) has a graded (resp. \(\rho\)-graded) augmentation, then any front diagram of \(K\) has a graded (resp. \(\rho\)-graded) ruling. This is the converse of the result by Fuchs and was independently proved by D. Fuchs and T. Ishkhanov [Mosc. Math. J. 4, No. 3, 707–717 (2004; Zbl 1073.53106)]. As a consequence the author proves that if the Chekanov-Eliashberg DGA of \(K\) has a \(2\)-graded augmentation, then \(r(K)=0.\)

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57M25 Knots and links in the \(3\)-sphere (MSC2010)
53D10 Contact manifolds (general theory)
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