Pretzel knots with unknotting number one

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 diagonalization theorem (as applied by Greene), Nakanishi’s unknotting bounds from the Alexander module, and the correction terms introduced by Ozsváth and Szabó. 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.

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Published 9 April 2013