The Unapologetic Mathematician

Friday, Carnies, and Khovanov homology

The new Carnival of Mathematics post is up over at Science and Reason, continuing my efforts to become the blogosphere’s go-to guy for publicizing the Atlas project.

This afternoon in the graduate student seminar, Joshua Sussan broke down parts of the famous paper of Bernstein, Frenkel, and Khovanov that launched knot homology theories: A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak{sl}_2)$ via projective and Zuckerman functors. Yes, Zuckerman is my advisor, and yes this also ties back into the stuff I’ve been talking about with respect to $E_8$.

One particular bit of self-promotion on this point: often Khovanov homology — either this original representation-theoretic approach, a later sheaf-theoretic approach, or Khovanov and Rozansky’s combinatoral version — is called a categorification of the Jones polynomial, or of the bracket polynomial. I’ve mentioned the bracket before, specifically in relation to my talk on bracket extensions. In fact, Khovanov homology on tangles categorifies one of my bracket-extending functors: $F_{V_1,\cap}$, where $V_1$ is the standard $2$-dimensional representation of the $q$-deformed enveloping algebra $U_q(\mathfrak{sl}_2)$ and $\cap$ is the canonical pairing from $V_1\otimes V_1$ to the trivial representation.

Don’t get me wrong. Khovanov homology is a truly brilliant idea, but I hold out hope that there’s some other categorification that makes it clear what the topological content of the Kauffman bracket polynomial is.

April 6, 2007 - Posted by | Uncategorized