Journal of Knot Theory and Its Ramifications

In this paper we give a short survey of exotic smooth structures of 4-manifolds. Various methods of constructing exotic structures and their relation to “Shake-Sliceness” and the Zeeman conjecture is discussed.

It has already been rigorously established that if G is a finite group and α∈H⁴(G; ℤ), there is an associated 2+1-dimensional topological quantum field theory. In the case α=0, there is a Hopf algebra associated to the theory. We give an intuitive geometric solution to the Yang-Baxter equation for and go on to show that is a modular Hopf algebra. We also prove a structure theorem concerning the Verlinde algebra which is seen to correspond to an algebra structure on the space of finite dimensional representations of . A knowledge of this structure aids the development of the skein theory for the link invariants of Reshetikhin and Turaev which are associated to .

We study link-homotopy classes of links in the three sphere using reduced groups endowed with peripheral structures derived from meridian-longitude pairs. Two types of peripheral structures are considered — Milnor’s original version (called “pre-peripheral structures” in Levine’s terminology) and Levine’s refinement (called simply “peripheral structures”). We show here that pre-peripheral structures are not strong enough to classify links up to link-homotopy, and that Levine’s peripheral structures, although strong enough to distinguish those classes not distinguished by pre-peripheral structures, are also in all likelihood not strong enough to distinguish all link-homotopy classes. Following Levine’s classification program, we compare structure-preserving and realizable automorphisms, using an obstruction-theoretic approach suggested by work of Habegger and Lin. We find that these automorphism groups are in general different, so that a more complex program for classification by structured groups is required.

The coincidence of two different presentations of Witten 3-manifold invariants is proved. One of them, invented by Reshetikhin and Turaev, is based on the surgery presentation a of 3-manifold and the representation theory of quantum groups; another one, invented by Kohno and Crane and, in slightly different language by Kontsevich, is based on a Heegaard decomposition of a 3-manifold and representations of the Teichmuller group, arising in conformal field theory. The explicit formula for the matrix elements of generators of the Teichmuller group in the space of conformal blocks in the SU(2)_k, WZNW-model is given,using the Jones polynomial of certain links.

Using techniques from [14] we construct topological quantum field theories using an algebraic model of a homotopy 2-type as initial data.