Centrality and the KRH Invariant

TL;DR

This paper explores the categorical structures underlying quantum invariants of knots and 3-manifolds, focusing on Hopf algebras and their role in defining and generalizing these invariants.

Contribution

It provides an abstract framework connecting Hopf algebra properties with topological invariants, extending to virtual knots and updating previous work.

Findings

Hopf algebra images of knots lie in the center of the algebra

Axiomatic properties of quasi-triangular Hopf algebras relate to topology via a functor

Invariants extend to rotational virtual knots

Abstract

The purpose of this paper is to discuss the categorical structure for a method of defining quantum invariants of knots, links and three-manifolds. These invariants can be defined in terms of right integrals on certain Hopf algebras. We call such an invariant of 3-manifolds a Hennings invariant. The work reported in this paper has its background in previous work of the authors. The present paper gives an abstract description of these structures and shows how the Hopf algebraic image of a knot lies in the center of the corresponding Hopf algebra. The paper also shows how all the axiomatic properties of a quasi-triangular Hopf algebra are involved in the topology via a functor from the Tangle Category to the Diagrammatic Category of a Hopf Algebra. The invariants described in this paper generalize to invariants of rotational virtual knots. The contents of this paper are an update of the original 1998 version published in JKTR.