# Link Crossing Number is NP-hard

## TL;DR

This paper proves that calculating the crossing number of a link is an NP-hard problem, and for some weaker equivalence notions, it is NP-complete, highlighting computational complexity challenges in knot theory.

## Contribution

It establishes the NP-hardness of the link crossing number problem and NP-completeness for certain weaker link equivalence notions, advancing complexity understanding in knot theory.

## Key findings

- Link crossing number determination is NP-hard.
- Weak link equivalence problems are NP-complete.
- Highlights computational complexity in knot theory.

## Abstract

We show that determining the crossing number of a link is NP-hard. For some weaker notions of link equivalence, we also show NP-completeness.