Algebraic & Geometric Topology

Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

Michael B Henry and Dan Rutherford

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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