Volume 13, issue 5 (2013)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra

Lenhard Ng and Daniel Rutherford

Algebraic & Geometric Topology 13 (2013) 3047–3097
Abstract

We develop a close relation between satellites of Legendrian knots in 3 and the Chekanov–Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in 3 and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We derive several consequences of this result, notably that the question of existence of ungraded finite-dimensional representations for the DGA of a Legendrian knot depends only on the topological type and Thurston–Bennequin number of the knot.

Keywords
Legendrian knot, Legendrian contact homology, normal ruling, satellite
Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 53D42, 57M25
References
Publication
Received: 15 June 2012
Revised: 8 April 2013
Accepted: 8 April 2013
Published: 1 August 2013
Authors
Lenhard Ng
Department of Mathematics
Duke University
Box 90320
Durham, NC 27708-0320
USA
http://www.math.duke.edu/~ng/
Daniel Rutherford
Department of Mathematics
University of Arkansas
301 SCEN
1 University of Arkansas
Fayetteville, AR 72701
USA