Wigner symbols and combinatorial invariants of three-manifolds with boundary (Q1583870)

The authors extend both the Ponzano-Regge function and the Turaev-Viro invariant to compact 3-dimensional simplicial PL-manifolds with non-empty boundaries. In their approach the presence of a 2-dimensional boundary is taken into account through the introduction in the state sum \(Z[(M, \partial M)]\) of a Wigner \(3jm\) symbol associated with each triangle lying in the boundary itself. The results established for the classical SU(2)-invariant \(Z[(M, \partial M)]\) are extended to the case of the quantized enveloping algebra \(U_q(sl(2,C))\) (\(q\) a root of unit). The resulting quantum invariant \(Z_q[(M, \partial M)]\) is the counterpart of the Turaev-Viro invariant for a closed 3-dimensional PL-manifold.