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

Ng, Lenhard¹ ; Rutherford, Daniel²

¹Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, USA
²Department of Mathematics, University of Arkansas, 301 SCEN, 1 University of Arkansas, Fayetteville, AR 72701, USA

Algebraic and Geometric Topology, Tome 13 (2013) no. 5, pp. 3047-3097

Résumé

We develop a close relation between satellites of Legendrian knots in ℝ3 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 ℝ3 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.

Détail

DOI : 10.2140/agt.2013.13.3047

Classification : 57R17, 53D42, 57M25
Keywords: Legendrian knot, Legendrian contact homology, normal ruling, satellite

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