# Coloring invariants of knots and links are often intractable

## TL;DR

This paper proves that counting certain group homomorphisms related to knot diagrams is computationally intractable, specifically #P-hard, extending previous results to braid group actions.

## Contribution

It establishes the #P-hardness of counting homomorphisms from knot groups to nonabelian simple groups, using braid group actions, advancing the understanding of computational complexity in knot invariants.

## Key findings

- Counting homomorphisms is #P-hard for nonabelian simple groups.
- The method uses braid group actions instead of mapping class groups.
- Extends previous complexity results to a broader class of knot-related problems.

## Abstract

Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms $\pi_1(S^3\setminus K) \to G$ that send meridians of $K$ to $C$ is almost parsimoniously $\mathsf{\#P}$-complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology 3-spheres to $G$ is almost parsimoniously $\mathsf{\#P}$-complete. Where we previously used mapping class groups actions on closed, unmarked surfaces, we now use braid group actions.