Modified 6j-symbols and 3-manifold invariants

Abstract

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in [6], lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects. Dedicated to Jose Maria Montesinos on the occasion of his 65th birthday

Some remarks on the unrolled quantum group of sl(2)

“…This category has been used to construct quantum link and 3-manifold invariants in several papers [2,4,8,10,22,24,25]. These 3-manifold invariants have powerful new properties, including asymptotic behavior related to the Volume Conjecture and novel quantum representation of mapping class groups (see [4,8]).…”

The trace on projective representations of quantum groups

“…The motivation of this paper is to provide the underpinnings for the construction of topological invariants. With this in mind, in this subsection, we recall the notion of re-normalized colored ribbon graph invariants introduced and studied in [19,17,20,21,22]. This subsection is independent of the rest of the paper.…”

“…Let F : Gr i C → C be the Reshetikhin-Turaev k-linear functor (see [21]). If C is a ribbon category, the functor on planer graphs F : Gr 2 C → C extends to the functor on spatial graphs F : Gr 3 C → C .…”

“…The notion of generically G-semi-simple categories appears in [21,19] through the following generalization of fusion categories (in particular, fusion categories satisfy the following definition when G is the trivial group, X = ∅ and d = b = qdim C is the quantum dimension):…”

“…A (modified) 6j-symbol is the value of a tetrahedron under F . These 6j-symbols are the elementary algebraic ingredients of a re-normalized Turaev-Viro type invariant of 3-manifolds defined by state sums on triangulations ( [21,19]). Example 2: Right trace.…”