Two-bridge knots with unknotting number one

Kanenobu, Taizo; Murakami, Hitoshi

doi:10.1090/S0002-9939-1986-0857949-8

Two-bridge knots with unknotting number one
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by Taizo Kanenobu and Hitoshi Murakami

Proc. Amer. Math. Soc. 98 (1986), 499-502

DOI: https://doi.org/10.1090/S0002-9939-1986-0857949-8

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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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