Handlebody-knot invariants derived from unimodular Hopf algebras

Abstract

To systematically construct invariants of handlebody-links, we give a new presentation of the braided tensor category [Formula: see text] of handlebody-tangles by generators and relations, and prove that given what we call a quantum-commutative quantum-symmetric algebra A in an arbitrary braided tensor category [Formula: see text], there arises a braided tensor functor [Formula: see text], which gives rise to a desired invariant. Some properties of the invariants and explicit computational results are shown especially when A is a finite-dimensional unimodular Hopf algebra, which is naturally regarded as a quantum-commutative quantum-symmetric algebra in the braided tensor category [Formula: see text] of Yetter–Drinfeld modules.