Algebraic & Geometric Topology

Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

Michael B Henry and Dan Rutherford

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Abstract

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.

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3323-3353.

Dates
Received: 25 July 2014
Revised: 10 April 2015
Accepted: 15 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841069

Digital Object Identifier
doi:10.2140/agt.2015.15.3323

Mathematical Reviews number (MathSciNet)
MR3450763

Zentralblatt MATH identifier
1334.57025

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 53D40: Floer homology and cohomology, symplectic aspects

Keywords
invariants Legendrian knots augmentations Morse complex sequences generating families differential graded algebra Legendrian isotopy contact structure normal ruling

Citation

Henry, Michael B; Rutherford, Dan. Equivalence classes of augmentations and Morse complex sequences of Legendrian knots. Algebr. Geom. Topol. 15 (2015), no. 6, 3323--3353. doi:10.2140/agt.2015.15.3323. https://projecteuclid.org/euclid.agt/1510841069


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