The Correspondence between Augmentations and Rulings for Legendrian Knots
نویسنده
چکیده
We strengthen the link between holomorphic and generatingfunction invariants of Legendrian knots by establishing a formula relating the number of augmentations of a knot’s contact homology to the complete ruling invariant of Chekanov and Pushkar.
منابع مشابه
Satellites of Legendrian Knots and Representations of the Chekanov–eliashberg Algebra
We study satellites of Legendrian knots in R 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 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...
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Each ruling of a Legendrian link can be naturally treated as a surface. For knots, the ruling is 2–graded if and only if the surface is orientable. For 2–graded rulings of homogeneous (in particular, alternating) knots, we prove that the genus of this surface is at most the genus of the knot. While this is not true in general, we do prove that the canonical genus (a.k.a. diagram genus) of any k...
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We show that if a Legendrian knot in standard contact R3 possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology is isomorphic to singular homology groups arising from the generating family. We discuss the relationship between normal rulings, augmentations, and generating families. In particular, we provid...
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We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R3. This ‘homotopy cardinality’ is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.
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