Mathematics Colloquia and Seminars

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The three classical invariants of a Legendrian link are its topological knot type, rotation number, and Bennequin number. Within any given topological knot type the Bennequin number can be arbitrarilly negative but is bounded above by certain topological invariants of the link. One such upper bound comes from the Kauffman polynomial. A ruling of a Legendrian link is a way of decomposing its front diagram into a link of Legendrian unknots by resolving crossings in an allowable fashion. Fuchs introduced the notion of a ruling and together with Ishkanov showed that for knots the existence of a ruling is equivalent to the existence of an augmentation on the Chekanov-Eliashberg DGA (a non-classical invariant). I will discuss recent work showing that (as conjectured by Fuchs) a third equivalent condition is the sharpness of the Kauffman polynomial estimate for the Bennequin number. This is done by realizing the number of rulings of a link as coefficients of the Kauffman polynomial. If time permits I will discuss analogous results regarding the HOMFLY polynomial and oriented rulings.