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Caitlin Leverson : Legendrian Knots, Augmentations, and Rulings

Given a contact structure (a plane field) on R^3, one can define a Legendrian knot to be an embedding of the circle such that the embedding is everywhere tangent to the plane field. Surgery along such a knot gives a way to construct new manifolds and so there is interest in classifying Legendrian knots. This turns out to be a finer classification than that of topological knots -- there are many different Legendrian unknots. Given a Legendrian knot, one can associate the Chekanov-Eliashberg differential graded algebra (DGA) generated by the crossings and then find augmentations of this DGA much like those in your standard algebraic topology course. This talk will give an overview of the relationships Joshua Sabloff and Dmitry Fuchs gave between such rulings and augmentations and how it relates to my current work.

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