Quasi-Quantum Groups, Knots, Three-Manifolds, and Topological Field Theory

TL;DR

This paper develops invariants for knots, links, and three-manifolds using quasi-Hopf algebras, connecting algebraic structures with topological quantum field theories, especially those related to Dijkgraaf-Witten models.

Contribution

It introduces a method to derive topological invariants from quasi-quantum groups, linking algebraic and topological quantum field theories in a novel way.

Findings

Invariants of knots and links from quasi-Hopf algebras.

Extension of invariants to three-manifolds via surgery.

Equivalence with Dijkgraaf-Witten topological field theory.

Abstract

We show how to construct, starting from a quasi-Hopf algebra, or quasi-quantum group, invariants of knots and links. In some cases, these invariants give rise to invariants of the three-manifolds obtained by surgery along these links. This happens for a finite-dimensional quasi-quantum group, whose definition involves a finite group $G$, and a 3-cocycle $\om$, which was first studied by Dijkgraaf, Pasquier and Roche. We treat this example in more detail, and argue that in this case the invariants agree with the partition function of the topological field theory of Dijkgraaf and Witten depending on the same data $G, \,\om$.