Modified 6j-Symbols and 3-Manifold Invariants

TL;DR

This paper introduces modified 6j-symbols and new state sum invariants for 3-manifolds derived from tensor categories, especially those that are pivotal but not ribbon or semi-simple, expanding the scope of quantum invariants.

Contribution

It develops a framework for modified 6j-symbols and 3-manifold invariants from tensor categories that are pivotal but not ribbon or semi-simple, with examples where standard invariants vanish.

Findings

Standard Turaev-Viro invariants can vanish for certain categories.

Modified invariants can be non-zero even when standard ones are zero.

Categories with infinite simple objects can produce non-trivial invariants.

Abstract

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] 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.