SimCenter and Department of Mathematics Colloquium
Friday, February 19, 2021 at 2:00 PM
Virtual Knot Theory
Louis H. Kauffman (Department of Mathematics, Statistics, and Computer Science – University of Illinois at Chicago)
Abstract: 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 talk 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 behaviors 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. This talk will be self-contained and accessible to undergraduate students and researchers outside of knot theory and algebraic topology.
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