# On the complexity of torus knot recognition

## TL;DR

This paper investigates the computational complexity of recognizing torus knots and related knot types, showing they are in NP and co-NP under certain assumptions, with implications for knot detection algorithms.

## Contribution

It establishes the complexity class membership of recognizing torus and satellite knots, advancing understanding of knot recognition problems under the generalized Riemann hypothesis.

## Key findings

- Torus knot recognition is in NP ∩ co-NP assuming GRH.
- Satellite knot detection is in NP under GRH.
- Cabled and composite knot detection are unconditionally in NP.

## Abstract

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in ${\sf NP}$ under the same assumption, and that cabled knot detection and composite knot detection are unconditionally in ${\sf NP}$. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness.