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Two-bridge knots with unknotting number one

Authors: Taizo Kanenobu and Hitoshi Murakami
Journal: Proc. Amer. Math. Soc. 98 (1986), 499-502
MSC: Primary 57M25
MathSciNet review: 857949
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Abstract: We determine all two-bridge knots with unknotting number one. In fact we prove that a two-bridge knot has unknotting number one iff there exist positive integers $ p$, $ m$, and $ n$ such that $ (m,n) = 1$ and $ 2mn = p \pm 1$, and it is equivalent to $ S(p,2{n^2})$ in Schubert's notation. It is also shown that it can be expressed as $ C(a,{a_1},{a_2}, \ldots ,{a_k}, \pm 2, - {a_k}, \ldots , - {a_2}, - {a_1})$ using Conway's notation.

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Keywords: Two-bridge knot, unknotting number, lens space, Dehn surgery
Article copyright: © Copyright 1986 American Mathematical Society

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