Louis Kauffman, Virtual Knot Theory and Khovanov Homology (lecture 3)

This course will study virtual knot theory, invariants of virtual knots and links and particularly it will concentrate on Khovanov homology for virtuals with applications.

Classical knot theory is the study of embeddings of the circle (knots) and of collections of circles (links) into three dimensional space (usually R^3 or S^3 — Euclidean three space or the three dimensional sphere). Since the 1920’s and the work of Alexander, Briggs and Reidemeister, the problem of isotopy of embeddings of knots and links in R^3 can be expressed in terms of planar diagrams of the knots and links. Such diagrams are 4-valent plane graphs with extra structure showing how to weave the knot or link in three dimensional space. It is of interest to have diagrammatic theories for knots and links in other three manifolds. Virtual knot theory studies knots and links embedded in thickened surfaces up to handle stabilization. One uses embeddings in S_g x I where S_g is an orientable surface of genus g and I is the unit interval. Stabilization means that two embeddings that can be related by adding or subtracting handles from the surface are taken to be equivalent. One is interested in the least genus that supports a given virtual knot or link. There is a diagrammatic theory for virtual knots. This course will begin with that diagrammatic theory, explaining how it captures the three manifold topology and how one can define combinatorial invariants such as the Jones polynomial for virtual knots. Just as in graph theory, where phenomena change when one leaves the category of planar graphs, there are distinct differences between the behaviours of classical knots and higher genus virtual knots. In particular there are infinitely many virtual knots with unit Jones polynomial. No such examples are known in the classical domain.

Khovanov homology is a remarkable generalization of the bracket model of the Jones polynomial to a homology theory that is a knot and link invariant. We will describe Khovanov homology for classical knots and then generalize it to virtual knots in the manner of Vassily Manturov, after preparing sufficient background. Our work will include generalizations of the Lee Homology and the Rasmussen Invariant. If time permits, we shall discuss the Lipshitz-Sarkar stable homotopy generalization of Khovanov homology and the work of Kauffman, Nikonov and Ogasa on these homotopy constructions in virtual knot theory. This course will be self-contained and accessible to students and researchers outside of knot theory and algebraic topology.

Sorry, the video of Lecture 2 is missed due to technical issues