Colloquium: A new method to distinguish Legendrian knots

Speaker: Ivan Dynnikov, Steklov Mathematical Institute, Moscow

Abstract: The talk is based on a joint ongoing work with Maxim Prasolov. Our main objects of interest are rectangular diagrams of knots, which are a promising tool in knot theory. They appear to be intimately related to Legendrian knots. Whereas the problem of comparing topological types of two knots is solvable both theoretically and---for a small number of crossings---practically, there is no regular method to decide whether two Legendrian knots having the same topological type and the same classical invariants are equivalent or not. A lot is known in particular cases, but there remain open questions already for knots with just six crossings.

Recently we extended the formalism of rectangular diagrams to representation of surfaces. This formalism turned out to work nicely for Giroux' convex surfaces. The latter are very useful for distinguishing contact structures and Legendrian knots. By using our formalism and properties of Giroux's convex surfaces we introduced a combinatorial method to distinguish Legendrian knots, which I will overview after introducing the basic notions of the subject like rectangular diagram or Legendrian knot.