Observables in the Turaev-Viro and Crane-Yetter models

Abstract

We define an invariant of graphs embedded in a 3-manifold and a partition function for 2-complexes embedded in a triangulated 4-manifold by specifying the values of variables in the Turaev-Viro and Crane-yetter state-sum models. In the case of the three-dimensional invariant, we prove a duality formula relating its Fourier transform to another invariant defined via the colored Jones polynomial. In the case of the four-dimensional partition function, we give a formula for it in terms of a regular neighborhood of the 2-complex and the signature of its complement. Some examples are computed which show that the partition function determines an invariant which can detect non locally-flat surfaces in a 4-manifold.