Mathematics Colloquia and Seminars

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In the early 80's Bennequin proved that the self-linking number of a transverse knot in the standard contact structure on S^3 was bounded above by minus the Euler characteristic of any Seifert surface for the knot. Eliashberg later proved the same bound in any tight contact manifold. It has been know for quite some time now that this bound is not optimal for many knot types. It turns out there is an elegant interaction between the optimality of the Bennequin inequality for fibered knots and Giroux's work on the relation between open books and contact structures. In this talk I will explain this interaction and give a precise characterization of when the Bennequin bound is optimal for fibered knots. However the talk will focus on some corollaries of this result, including applications to braid theory and a proof that "transverse knots classify contact structures". Most of the work described in this talk is joint with Jeremy Van Horn-Morris.