Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra

We develop a close relation between satellites of Legendrian knots in $ℝ^{3}$ 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 $ℝ^{3}$ 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.

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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

Volume 13, issue 5 (2013)

Lenhard Ng and Daniel Rutherford

Abstract