#### Volume 15, issue 6 (2015)

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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
##### Abstract

Let $L$ be a Legendrian knot in ${ℝ}^{3}$ 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.

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##### Keywords
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
##### Publication
Revised: 10 April 2015
Accepted: 15 April 2015
Published: 12 January 2016
##### Authors
 Michael B Henry Department of Mathematics Siena College 515 Loudon Road Loudonville, NY 12211 USA Dan Rutherford Department of Mathematics Ball State University 2000 W University Ave Muncie, IN 47306 USA