Room P3.10, Mathematics Building

Louis Kauffman, University of Illinois, Chicago
Virtual Knot Theory and Oriented Extensions of the Jones Polynomial

The original Jones polynomial for classical knots depends only weakly on the orientation of the knot or link to which it is applied. Careful examination of the orientations assigned to states in an oriented bracket polynomial model for the Jones polynomial reveals that there is much more topological information available from orientation when the knot or link is embedded in a thickened surface or regarded as a virtual link. This talk will discuss a large scale generalization that we call the extended bracket polynomial (taking polynomial and graphical values) and the arrow polynomial (joint work with Heather Dye) taking polynomial values with infinitely many new variables. The arrow polynomial is related to the Miyazawa polynomials for virtual knots. The talk will discuss applications of these new invariants to finding virtual crossing number and genus of virtual knots, and we shall discuss extensions of Khovanov homology related to these invariants.