© | << < ? > >> | Dror Bar-Natan: Talks:

Homomorphic Expansions and w-Knots

Mathematics Colloquium, University of Western Ontario
Thursday February 25, 2010

Abstract. Even though little known, the notion of a "homomorphic expansion" is extremely general; it makes sense in the context of practically any algebraic structure, be it a group, or a group homomorphism, or a quandle, or a planar algebra, or a circuit algebra with unzip operations, or whatever.

Even though little known, w-knots make a cool generalization of ordinary knots. They contain ordinary knots and are contained in 2-knots in 4-space and are easier than the latter. They are a quotient of "virtual knots" and are easier then those.

My talk will be about these two notions, homomorphic expansions and w-knots, and about what happens when the two are put together. Lie algebras arise, and Lie groups, and the Kashiwara-Vergne statement, which is one of the deeper statements about the relationship between Lie groups and Lie algebras.

There are also u-knots, and v-knots, and f-knots, and other things which are not knots at all, and there are equally nifty things to say about homomorphic expansions for all those. But not today.

Handout. wKnots.html, wKnots.pdf