Modified 6j-symbols and 3-manifold invariants

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in Geer et al. (2009) [14], 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.