Monday, February 23, 2004

Speaker: Mario Bourgoin (Brandeis)

Title: Virtual Knots and Beyond

Abstract: After a century of studying knots in space, Louis Kauffman introduced "virtual knots" as a generalization of knots to stabilized thickened closed surfaces. This generalization, inspired by the desire to make all Gauss codes represent links, comes with a theory of diagrams and extensions of the traditional invariants such as the group of the link and the Jones polynomial, and has seen rapid development in the past decade. At nearly the same time, Yu. V. Drobotukhina introduced "projective knots" as a generalization of knots to projective space, and defined for them a theory of diagrams in disks and a Jones polynomial. This area has had some recent developments with the introduction of an Alexander polynomial.

We introduce a common framework for both of these extensions as knots in stabilized oriented thickenings of (possibly non-oriented) closed surfaces. This framework is based on a Gauss code classification of all generic curves in stabilized (possibly non-oriented) closed surfaces. Also, we will present a theory of diagrams of these knots, and define a Jones polynomial for them. This extends the classical combinatorial theory of knots and links to a new class of three-manifolds, as well as clarifies the relationship between the concepts of writhe and height that come from classical knot theory.