State sum invariants of three manifolds from spherical multi-fusion categories

State-Sum Invariants of 4-Manifolds, I

Abstract: We provide, with proofs, a complete description of the authors’ construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of Re...

Spherical Tetrahedra and Invariants of 3-manifolds

Let Y be an oriented closed three-manifold and r a positive integer. The ReshetikhinTuraev invariant Z(Y, r) and Turaev-Viro invariant TV (Y, r) are three-manifold invariants that attempt to make rigorous the Hamiltonian formulation of quantum ChernSimons theory. Z(Y, r) is constructed using the R-matrix of the quantized enveloping algebra Uq(sl2) and Kirby moves, while TV (Y, r) is based on th...

Spherical 2-categories and 4-manifold Invariants

Three Manifolds and Graph Invariants

We show how the Turaev–Viro invariant can be understood within the framework of Chern–Simons theory with gauge group SU(2). We also describe a new invariant for certain class of graphs by interpreting the triangulation of a manifold as a graph consisiting of crossings and vertices with three lines. We further show, for S3 and RP 3, that the Turaev-Viro invariant is the square of the absolute va...

State Sum Invariants of Compact 3-manifolds with Boundary and 6j-symbols. *

We extend the combinatorial construction of invariants of smooth, compact closed 3-manifolds as given by Turaev and Viro to obtain invariants of 3-manifolds with boundary. The technique uses quantum 6j-symbols associated to the quantized universal enveloping algebra U q (s`(2; C)) (q a root of unity) to give a construction of a state sum for given triangulation. This state sum is then invariant...