Building Blocks in Turaev-Viro Theory

TL;DR

This paper investigates the structure of the Turaev-Viro partition function for 3-manifolds with boundary, revealing factorization properties and formulas for connected sums, with specific results for spheres and tori boundaries.

Contribution

It demonstrates the factorization of Z(M) for S^2 boundaries and derives a formula for connected sums, extending understanding of Turaev-Viro invariants.

Findings

Z(M) factorizes for S^2 boundaries into boundary-dependent and topology-dependent parts

Derived a formula for Z(M # N) for connected sums of manifolds

Factorization for T_g boundaries holds only in specific cases

Abstract

We study the form of the Turaev-Viro partition function Z(M) for different 3-manifolds with boundary. We show that for $S^2$ boundaries Z(M) factorizes into a term which contains the boundary dependence and another which depends only on the topology of the underlying manifold. From this follows easily the formula for the connected sum of two manifolds Z(M # N). For general $T_g$ boundaries this factorization holds only in a particular case.