Handlebody-knot invariants derived from unimodular Hopf algebras

TL;DR

This paper develops a general framework for constructing invariants of handlebody-knots using category theory and unimodular Hopf algebras, providing a new algebraic approach to topological invariants.

Contribution

It introduces a categorical method to derive handlebody-knot invariants from quantum-commutative quantum-symmetric algebras, focusing on unimodular Hopf algebras.

Findings

Constructed a category of handlebody-tangles with generators and relations.

Showed that invariants arise from quantum-commutative quantum-symmetric algebras.

Explored how unimodular Hopf algebras produce handlebody-knot invariants.

Abstract

A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a uniform method to construct invariants for handlebody-links. We introduce the category $\mathcal{T}$ of handlebody-tangles and present it by generators and relations. The result tells us that every functor on $\mathcal{T}$ that gives rise to invariants is derived from what we call a quantum-commutative quantum-symmetric algebra in the target category. The example of such algebras of our main concern is finite-dimensional unimodular Hopf algebras. We investigate how those Hopf algebras give rise to handlebody-knot invariants.