The combinatorial description of Witten's invariant of 3- manifolds and links which the investigator developed in his last two papers will be explored in a number of directions. The combinatorial technique will be extended to include knotted graphs. A reformulation of the invariant of trivalent graphs in terms of tetrahedral decompositions will be developed. The implications of this for invariants of the manifolds themselves will be explored, and a relationship sought with the parallel work of Viro and Turaev. A geometric interpretation of this tetrahedral formula will also be attempted, in close analogy with Regge and Ponzano's interpretation of spin networks. Connections will also be explored with invariants in other dimensions: the Donaldson invariants in four dimensions and surgery invariants for dimensions five and above. In short, the investigator will explore topological invariants inspired by quantum physics. He will examine their usefulness in purely topological settings. He will also explore the physical implications of the resulting topology.
