AKT-14

Algebraic Knot Theory

Department of Mathematics, University of Toronto, Spring 2014

I will be giving an Algebraic Knot Theory class at the University of Toronto in the spring semester of 2014, and this will be its home page.

Agenda. Understand "(u, v, and w knots) x (topology, combinatorics, low algebra, and high algebra)". Understand the promise and the difficulty of the not-yet-existent "Algebraic Knot Theory".

Description. There are many types of "knots", and many things to do with them. Specifically, we will talk about three master-types of knots - "u" knots which are just the usual knots we are used to seeing in 3-space, "w" knots which are 2-dimensional knots in 4-space, and "v" knots which are "virtual" knots, algebraic creatures that are not specifically embedded anywhere. Each of these master-types comes in several shades - there are plain uvw-knots, and uvw-tangles, and uvw-braids, and uvw-knotted-graphs, and there is a rich world of operations that can be applied to these, and a rich world of properties and questions to ask. And then there is the projectivization machine, which converts all these to combinatorial objects, and then algebraic (of two classes, low and high). The "Fundamental Problem" is always to construct a good map (technically, a "homomorphic expansion") from topology to combinatorics/algebra, and in the cases we understand, the solution of the Fundamental Problem involves matters such as the Knizhnik–Zamolodchikov connection, topological quantum field theory and Feynman diagrams and configuration space integrals, Drinfel'd associators and the pentagons and hexagons of category theory, and deep Lie theory. Everything we will mention, but not everything we will cover as deeply as I'd like to.

References.

• Introduction to Vassiliev Knot Invariants, by S. Chmutov, S. Duzhin, and J. Mostovoy, Cambridge University Press, Cambridge UK, 2012.
• My own papers and talks and classes as can be found at http://www.math.toronto.edu/~drorbn/.

Prerequisites. A high level of comfort with vector spaces and algebras and things you can do with them - quotients, tensor products, duals, etc. Basic topology - fundamental groups and van Kampen and rarely a bit of homology. Basic differential geometry - especially differential forms and Stokes' formula. A basic knowledge of Lie groups and algebras and their representations and an appreciation of their value. You will manage with some of that missing, but the more you will be missing the more lost you will be at times.