CHAPTER 2 Seiferters and Seifert Surgery Network 2.1. Seiferters for Seifert surgeries Let K be a knot in the interior of a 3–manifold X with a tubular neighborhood N(K). Let γ be a slope, i.e. the isotopy class of essential unoriented simple closed curves on ∂N(K). Denote by X(K, γ) the 3–manifold obtained from X by γ– surgery on K. If X S3, we choose a preferred meridian–longitude pair (μ, λ) of N(K) S3. Then the set of slopes γ is parametrized by Q { 1 0 }, by the correspondence [a representative of γ with an orientation] = q[μ] + p[λ] r = q p in H1(∂N(K)). (The ambiguity of the choice of the orientation of the representative disappears after taking ratio.) A simple closed curve on ∂N(K) is called a + curve if it represents q[μ]+p[λ]. We often write X(K r) for X(K γ). In particular, if X = S3, then X(K r) is denoted by K(r). A Dehn surgery (K, r) is called a Seifert fibered surgery if the resulting manifold K(r) is a Seifert fiber space. It has been conjectured that if (K, r) is a Seifert fibered surgery, then r is an integer except when K is a torus knot or a cable of a torus knot. The conjecture is verified for several cases [14, 13, 16, 37, 63]. In [65] the second and the third authors proved that many Seifert fibered surgeries (K, r) have the following interesting property. Property 2.1. There exists a knot c in S3 N(K) with the following proper- ties. (1) c is unknotted in S3. (2) c becomes a Seifert fiber in the resulting Seifert fiber space K(r). We call such a curve c a seiferter for a Seifert fibered surgery (K, r). As shown in Theorem 2.2 below the existence of seiferters implies the positive solution to the conjecture. Hence, in this paper we only consider integral Seifert surgeries (K, m), where m Z. Theorem 2.2 ([65]). Suppose that a Seifert fibered surgery (K, r) has a seifer- ter. Then r is an integer or K is a torus knot or a cable of a torus knot. All known Seifert surgeries except for one infinite family obtained by Teragaito [80] have seiferters in general they have more than one seiferter. In the present paper, we focus on another property of seiferters which enables us to relate Seifert surgeries. If we have a seiferter c for a Seifert fibered surgery (K, m) where m Z, then by twisting K p times along the seiferter c, we obtain another knot Kp. Let mp be the slope on ∂N(Kp) which is the image of m after 5
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