Representation Theory and Mathematical Physics Seminar: Introduction to Virtual Knot Theory

Seminar | December 6 | 2-3 p.m. | 736 Evans Hall | Note change in time

 Louis H Kauffman, University of Illinois at Chicago

 Department of Mathematics

Virtual knot theory is a generalization of classical knot theory that studies stabilized knots and links in thickened surfaces. Two knots (links) in thickened surfaces are said to be stably equivalent if they can be obtained one from another by a finite sequence of ambient isotopies along with surgeries on their complements (that can change of genus of the embedding surface). There is a diagrammatic theory that captures stable equivalence. One adds virtual crossings (neither over nor under) and rules for handling them that generalize the Reidemeister moves. Then virtual knots can be studied using strictly planar diagrams. This means that one has access to both a rich background of combinatorial topology and the three dimensional topology of the thickened surfaces. This talk will discuss the basic definitions for virtual knot theory and the construction of a number of invariants of interest, including the Jones polynomial, the arrow polynomial and the affine index polynomial, Khovanov homology and relations with virtual knot cobordism. We will discuss how quantum link invariants extend to virtual knot theory and we will attempt to discuss how virtual knot theory could or should be related to physics and quantum information theory.