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Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra
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We study satellites of Legendrian knots in R^3 and their relation to 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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Legendrian DGA Representations and the Colored Kauffman Polynomial
For every Legendrian knot, ungraded n-dimensional representation numbers of its contact homology DGA equal the n-colored Kauffman polynomial specialized at a^{-1}=0.
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