The unbearable hardness of unknotting

TL;DR

This paper establishes the NP-hardness of several fundamental problems in knot theory and link detection, including unknotting and computing various invariants, highlighting their computational complexity.

Contribution

It proves that deciding if a knot diagram can be untangled with a limited number of Reidemeister moves is NP-hard, along with other link-related problems.

Findings

Deciding unknotting with limited moves is NP-hard.

Detecting trivial sublinks in links is NP-hard.

Computing unlinking number and certain 4D invariants is NP-hard.

Abstract

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the $4$-ball Euler characteristic, the slicing number, and the $4$-dimensional clasp number).