# Quantized representations of knot groups

## TL;DR

This paper introduces a novel non-commutative framework for knot group representations using braided Hopf algebras, leading to new quantum invariants and character varieties that extend classical concepts.

## Contribution

It develops a braided Hopf algebra approach to generalize representation and character varieties of knot groups, creating new quantum invariants and algebraic structures.

## Key findings

- Constructed a non-commutative knot invariant as a module with coadjoint action.
- Defined a quantum character variety as coinvariants, offering an alternative to skein modules.
- Provided explicit examples for simple knots and links.

## Abstract

We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construction works under the assumption that the algebra is braided commutative. The resulting knot invariant is a module with a coadjoint action. Taking the coinvariants yields a new quantum character variety that may be thought of as an alternative to the skein module. We give concrete examples for a few of the simplest knots and links.