Abstract. The right objects of study in algebraic topology are not spaces, but rather, "spaces and maps between them". In a similar spirit I will argue that the right things to study in knot theory are not knots, but rather, "knotted trivalent graphs", as in the world of knotted trivalent graphs (and the basic operations between them) many interesting properties of honest knots become "definable". Thus I find myself again studying the good old Kontsevich integral - the best example I know of an algebraic knot theory - but my perspective this time is completely different.
This talk will follow a talk I gave in Copenhagen in October, 2008. See the transparencies at http://www.math.toronto.edu/~drorbn/Talks/Copenhagen-081009.
Abstract. Many fundamental concepts in mathematics are the results of forgetting something about concepts that are even more fundamental. Many binomial identities, for example, are the q=1 specializations of even more powerful "q-binomial" identities, whose real home is in some non-commutative universes. Likewise the Euler characteristic of a space is the result of forgetting something about its homology; it is a beautiful and powerful number, but the loss is noticed - homology is a much stronger concept.
Recently it has become clear through the work of Khovanov that many knot invariants, and perhaps many other algebraic objects, are in some sense the Euler characteristics of objects that are potentially much more interesting.
In my talk I will quickly review the easy and elegant Jones polynomial of knots (a "q-object" in itself!). Then I will display one complicated picture and discuss it at length. After some definitions and some intepretation, we will see how it realizes Khovanov's idea in a local manner, leading to some theoretical and computational advantages.
This talk will follow a talk I gave in Zurich in May, 2008. See the handouts LocalKh (source), and 10 Minutes on Homology (source).
Pictures. Image Gallery: Places: Bogota, February 2009.
"God created the knots, all else in topology is the work of mortals."
Leopold Kronecker (rephrased)