A combinatorial approach to topological quantum field theories and invariants of graphs

The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any Riemann surface with arbitrary coloured punctures.Dedicated to H. Araki and E. Lieb on the occasion of their 60^th birthdaysSupported by DFG, SFB 288 Differentialgeometrie und Quantenphysik