A polynomial upper bound on Reidemeister moves for each link type

TL;DR

This paper proves that for any link type, there exists a polynomial bound on the number of Reidemeister moves needed to relate two diagrams, leading to complexity results for link recognition.

Contribution

It establishes a polynomial upper bound on Reidemeister moves for each link type, enabling NP classification of link recognition problems.

Findings

Polynomial bounds on Reidemeister moves for all link types

Link recognition problem is in NP due to polynomial bounds

Explicit polynomial bounds calculated for various link classes

Abstract

For each link type $K$ in the 3-sphere, we show that there is a polynomial $p_K$ such that any two diagrams of $K$ with $c_1$ and $c_2$ crossings differ by at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves. As a consequence, the problem of recognising whether a given link diagram represents $K$ is in the complexity class NP and hence can be completed deterministically in exponential time. We calculate this polynomial $p_K$ explicitly for various classes of links.