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We provide a partial classification of the 3-strand pretzel knots $K =P(p,q,r)$ with unknotting number one. Following the classification by Kobayashiand Scharlemann-Thompson for all parameters odd, we treat the remainingfamilies with $r$ even. We discover that there are only four possiblesubfamilies which may satisfy $u(K) = 1$. These families are determined by thesum $p+q$ and their signature, and we resolve the problem in two of thesecases. Ingredients in our proofs include Donaldson's diagonalisation theorem(as applied by Greene), Nakanishi's unknotting bounds from the Alexandermodule, and the correction terms introduced by Ozsv\'ath and Szab\'o. Based onour results and the fact that the 2-bridge knots with unknotting number one arealready 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.

Pretzel Knots with Unknotting Number One