# Invariants of long knots

## TL;DR

This paper reviews the construction of invariants for long knots using monoidal categories, R-matrices, and Hopf algebras, extending universal invariants to broader algebraic frameworks.

## Contribution

It introduces a new universal long knot invariant associated with any Hopf algebra with invertible antipode, extending previous finite-dimensional and topological approaches.

## Key findings

- Constructs a universal invariant $Z_H(K)$ for long knots from Hopf algebras.
- Extends known invariants to infinite-dimensional Hopf algebras via the quantum double.
- Highlights the importance of monoidal categories and algebraic structures in knot invariants.

## Abstract

By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations and spans over sets, by introducing racks associated with pointed groups, we illustrate the construction and the importance of consideration of long knots. Else, by using the restricted dual of algebras and Drinfeld's quantum double construction, we show that to any Hopf algebra $H$ with invertible antipode, one can associate a universal long knot invariant $Z_H(K)$ taking its values in the convolution algebra $((D(H))^o)^*$ of the restricted dual Hopf algebra $(D(H))^o$ of the quantum double $D(H)$ of $H$. That extends the known constructions of universal invariants previously considered mostly either in the case of finite dimensional Hopf algebras or by using some topological completions.