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Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

Michael B Henry and Dan Rutherford

Algebraic & Geometric Topology 15 (2015) 3323–3353

Let L be a Legendrian knot in 3 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.

invariants, Legendrian knots, augmentations, Morse complex sequences, generating families, differential graded algebra, Legendrian isotopy, contact structure, normal ruling
Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 57M25, 53D40
Received: 25 July 2014
Revised: 10 April 2015
Accepted: 15 April 2015
Published: 12 January 2016
Michael B Henry
Department of Mathematics
Siena College
515 Loudon Road
Loudonville, NY 12211
Dan Rutherford
Department of Mathematics
Ball State University
2000 W University Ave
Muncie, IN 47306