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Product structures for Legendrian contact homology

Published online by Cambridge University Press:  02 December 2010

GOKHAN CIVAN ,
PAUL KOPROWSKI ,
JOHN ETNYRE ,
JOSHUA M. SABLOFF  and
ALDEN WALKER
GOKHAN CIVAN
Affiliation:
University of Maryland, College Park, MD 20742, U.S.A.
PAUL KOPROWSKI
Affiliation:
University of Maryland, College Park, MD 20742, U.S.A.
JOHN ETNYRE
Affiliation:
Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. e-mail: etnyre@math.gatech.edu
JOSHUA M. SABLOFF
Affiliation:
Haverford College, Haverford, PA 19041, U.S.A.
ALDEN WALKER
Affiliation:
California Institute of Technology, Pasadena, CA 91125, U.S.A.
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Abstract

Legendrian contact homology (LCH) is a powerful non-classical invariant of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and non-commutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products – and, more generally, an A∞ structure – on the linearized LCH. We apply the products and A∞ structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 291 - 311
Copyright
Copyright © Cambridge Philosophical Society 2010

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