Two-Bridge Knots with Unknotting Number One

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.