Legendrian and transverse knots

This workshop, sponsored by AIM and the NSF, concerns Legendrian and transverse knots in contact three-dimensional manifolds. The focus will be on the interplay between topological knot theory and contact topology. The study of knots in contact space goes back to the early 1980s when D. Bennequin used knots to establish the existence of exotic contact structures. Since the late 1990s this area has experienced a surge of activity, after the work of Yu. Chekanov and Ya. Eliashberg on the contact homology of Legendrian knots.

One of the main goals of the workshop is to achieve a better understanding of a somewhat mysterious combinatorial structure of front diagrams of Legendrian knots, the so-called, normal rulings. This structure is closely related to seemingly independent properties of Legendrian knots: the existence of generating families of functions, certain algebraic properties of the Chekanov--Eliashberg algebra (augmentations), and the sharpness of the estimates of the self-linking (Thurston--Bennequin) numbers of Legendrian knots. A second goal is to make progress in the study of transverse knots, in particular, by constructing transverse invariants of the contact homology type. Further goals include applying Legendrian knot theory to topological knot theory and finding applications of Legendrian knot theory to global singularity theory in the spirit of Chekanov and Pushkar’s recent proof of Arnold's four cusps conjecture.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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