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• W2035904864 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 (, n) 1 and 2mn = p ? 1, and it is equivalent to S(p, 2n2) in Schubert's notation. It is also shown that it can be expressed as C(a, al, a2,... ,ak, +2, -ak,... ,-a2, -al) using Conway's notation. Let K be a knot in a 3-sphere. An unknotting operation is an operation which changes the overcrossing and the undercrossing at a double point of a diagram of K. The unknotting number of K, denoted by u(K), is the minimum number of unknotting operations needed to deform a diagram of K into that of the trivial knot, where the minimum is taken over all diagrams of K. By a two-bridge knot S(p, q) we mean a knot which is characterized so that its double branched covering space is the lens space L(p, q), where p and q are coprime integers and p is odd and positive [3, 6, 11, 12]. (Thus we regard S(p, q) and its mirror image S(p, -q) as equivalent.) Let C(cl,C2,... Cr) be Conway's notation for a two-bridge knot. If the continued fraction 1 1 C, + -2 -C is equal to p/q, then C(cl,C2,. . . ,Cr) is equivalent to S(p,q) [3, 12]. In this paper we consider two-bridge knots with unknotting number one and determine them. In fact we prove THEOREM 1. Let K be a nontrivial two-bridge knot. Then the following three conditions are equivalent. (i) u(K) = 1. (ii) There exist an odd integer p (> 1) and coprime, positive integers m and n with 2mn = p ? 1 and K is equivalent to S(p, 2n2). (iii) K can be expressed as C(a, al, a2, . . , ak, ?2,-ak, *,-a2,-al). To prove the above theorem we use the following theorem due to M. Culler, C. McA. Gordon, J. Luecke, and P. B. Shalen [4, 5] (see also Theorem A in [13]). THEOREM 2 [5]. For a knot K, let K(a/b) be a 3-manifold obtained by (a/b)Dehn surgery along K, where a and b are coprime integers. If K is not a torus knot and ri1(K(a/b)) is cyclic, then Ibl < 1. PROOF OF THEOREM 1. (i)=(ii). It is known that if a nontrivial knot K has unknotting number one then its double branched covering space is K(p/ ? 2) Received by the editors May 17, 1985. 1980 Mathematics Subject Classification. Primary 57M25." @default.
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• W2035904864 title "Two-bridge knots with unknotting number one" @default.
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• W2035904864 doi "https://doi.org/10.1090/s0002-9939-1986-0857949-8" @default.
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