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We define the "blob complex" B_*(M, C) associated to an n-manifold M and a (suitable) n-category C. This is a simultaneous generalisation of two interesting gadgets. When n=1, M = S^1 and C is an algebra, the homology of the blob complex is the Hochschild homology of the algebra. On the other hand, the zero-th homology of the blob complex is the usual skein module of "pictures from C drawn on M". In this sense the blob complex is a "derived" version of a TQFT.
We can prove several interesting properties of the blob complex. It's a functorial construction, and diffeomorphisms of the manifold act on the complex. Moreover, there's an action of chains of diffeomorphisms as well. Thus for example on the torus we get not only an action of the mapping class group, but also a compatible action of rotations along rational slopes. There appears to be a good gluing formula, expressed in terms of A_∞ bimodules. (The details here are still being worked out.)
We hope to apply blob homology to tight contact structures (for n=3) and Khovanov homology (for n=4). In both theories exact triangles play an important role. These exact triangles don't interact well with the gluing structure of the usual TQFTs, however. One of our motivations for considering blob homology is to work around these difficulties.
The representation theory of a quantum group forms a planar algebra (equivalently, a spider or pivotal category). For U_q(sl_2) and U_q(sl_3), there are nice combinatorial models (that is, finite presentations by generators and relations) of the planar algebras. These are the Temperley-Lieb algebra, and Kuperberg's A_2 spider, respectively.
I've made some progress extending these ideas to treat all U_q(sl_n). It's easy now to find a set of generators for the planar algebra. The difficulty is in finding all the relations amongst them. We can understand these relations by thinking about the inclusion of SU(n) inside SU(n+1), and thence the way that representations of U_q(sl_{n+1}) break up as representations of U_q(sl_n). This theory of "branching" and "Gelfand-Tsetlin bases" can be described combinatorially in terms of the planar algebra, and using this we can lift relations from one level to the next. My Berkeley Ph.D. thesis (see below) gives the details. It remains a conjecture that the relations I describe are complete.
With Noah Snyder, I'm working on some computations, and a paper, about the Kashaev-Reshetikhin knot invariants.
We first explain details from the papers of Kashaev and Reshetikhin to give a fully rigorous construction of their new knot invariants. This invariant is a function on the space Hom(G_m(K), SL_2)/SL_2 where G_m(K) is the generalized knot group, and the action of SL_2 is by conjugation. We prove several basic results, for example, that the value of the function on the trivial point is |J_m(\zeta_m)|^2.
Further, we give the first computations of this invariant for nontrivial knots, based on a Mathematica package we've written. For several small 2-bridge knots we compute explicitly the entire Kashaev-Reshetikhin knot invariant at a third root of unity and at a fifth root of unity. We're planning to compute the knot invariants evaluated at the finite volume hyperbolic point for a larger collection of 2-bridge knots (again at a third and fifth root of unity).