Pretzel Knots with Unknotting Number One

TL;DR

This paper classifies 3-strand pretzel knots with unknotting number one, focusing on cases with even r, and identifies specific families that satisfy this property, using advanced topological tools.

Contribution

It extends the classification of pretzel knots with unknotting number one by analyzing cases with even r and identifying potential exceptions.

Findings

Only four subfamilies of pretzel knots may have unknotting number one.

The problem is resolved for two of these subfamilies.

Conjecture that only specific non-2-bridge pretzel knots have unknotting number one.

Abstract

We provide a partial classification of the 3-strand pretzel knots $K = P(p,q,r)$ with unknotting number one. Following the classification by Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the remaining families with $r$ even. We discover that there are only four possible subfamilies which may satisfy $u(K) = 1$. These families are determined by the sum $p+q$ and their signature, and we resolve the problem in two of these cases. Ingredients in our proofs include Donaldson's diagonalisation theorem (as applied by Greene), Nakanishi's unknotting bounds from the Alexander module, and the correction terms introduced by Ozsv\'ath and Szab\'o. Based on our results and the fact that the 2-bridge knots with unknotting number one are already classified, we conjecture that the only 3-strand pretzel knots $P(p,q,r)$ with unknotting number one that are not 2-bridge knots are $P(3,-3,2)$ and its reflection.