Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

Henry, Michael B¹ ; Rutherford, Dan²

¹Department of Mathematics, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA
²Department of Mathematics, Ball State University, 2000 W University Ave, Muncie, IN 47306, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3323-3353

Résumé

Let L be a Legendrian knot in ℝ3 with the standard contact structure. In earlier work of Henry, a map was constructed from equivalence classes of Morse complex sequences for L, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of L. Moreover, this map was shown to be a surjection. We show that this correspondence is, in fact, a bijection. As a corollary, homotopic augmentations determine the same graded normal ruling of L and have isomorphic linearized contact homology groups. A second corollary states that the count of equivalence classes of Morse complex sequences of a Legendrian knot is a Legendrian isotopy invariant.

Détail

DOI : 10.2140/agt.2015.15.3323

Classification : 57R17, 57M25, 53D40
Keywords: invariants, Legendrian knots, augmentations, Morse complex sequences, generating families, differential graded algebra, Legendrian isotopy, contact structure, normal ruling

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