Pretzel Knots with Unknotting Number One

Details Pretzel Knots with Unknotting Number One Pretzel Knots with Unknotting Number One

2011-09-21

arxiv.org/abs/1109.4560v2

Mathematics

Geometric Topology

26 pages, 11 figures; includes additional cases, mistake in previous version fixed

Scientific paper

We provide a partial classification of all 3-strand pretzel knots $P(p,q,r)$ with unknotting number one. Following the classification by both Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the families with $r = 2m$. We first determine that there are only four possible families of 3-stranded pretzel knots (excluding 2-bridge knots) with $r$ even and which may have unknotting number one. These families are determined by the sum $p+q$ as well as the sign of a certain Goeritz matrix $G(K)$, and we solve the problem in two of these families. Ingredients in our proofs include Donaldon's diagonalisation theorem (and Greene's strengthening thereof), Nakanishi's unknotting bounds from the Alexander module, and the correction terms introduced by Ozsv\'ath and Szab\'o. Based on our results, we conjecture that the only 3-stranded pretzel knots $P(p,q,r)$ with unknotting number one that are not 2-bridge knots are $P(3,-3,2)$ and its reflection.

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