A polynomial upper bound on Reidemeister moves

TL;DR

This paper establishes a polynomial upper bound on the number of Reidemeister moves needed to simplify unknot diagrams and split links, providing a significant step in understanding knot diagram transformations.

Contribution

It introduces a polynomial upper bound on Reidemeister moves for unknot and split link diagrams, improving previous exponential bounds.

Findings

Unknot diagrams with c crossings can be simplified with at most (236 c)^{11} moves.

Sequences of moves maintain diagrams with at most (7 c)^2 crossings.

Polynomial bounds apply to split links as well.

Abstract

We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar theorem for split links, which provides a polynomial upper bound on the number of Reidemeister moves required to transform a diagram of the link into a disconnected diagram.