CHAPTER 1 Introduction We express an r–Dehn surgery on a knot K in S3 as the pair (K, r), and the resulting manifold as K(r). If K(r) is a Seifert fiber space which is allowed to have a degenerate fiber (i.e. an exceptional fiber of index zero), then (K, r) is called a Seifert surgery. For a torus knot K, since the exterior S3 intN(K) is Seifert fibered, (K, r) is a Seifert surgery for all slopes r. On the other hand, if K is a hyperbolic knot, then (K, r) is a Seifert surgery only for finitely many slopes r by Thurston’s hyperbolic Dehn surgery theorem [81, 82, 4, 75, 10]. Hence, Seifert surgeries are sparse among all Dehn surgeries. Although there have been several methods of construct- ing Seifert surgeries [5, 8, 12, 14, 18, 26, 80], no attention is paid to how Seifert surgeries are related to each other, and particularly whether Seifert surgeries on hy- perbolic knots and torus knots are related in some way. We relate Seifert surgeries by twisting operations which deform Seifert surgeries to other Seifert surgeries, and form a network of Seifert surgeries, called a Seifert Surgery Network. A subnetwork formed on the set of Seifert surgeries on torus knots is a distinguished subcomplex and denoted by T . Then, finding a path from a Seifert surgery to T is tracking down the origin of the surgery in T . Now let us take a closer look at the content of this paper. The definition of the Seifert Surgery Network is based on the notion of a seiferter for a Seifert surgery (K, r). If a knot c in S3 K is trivial in S3 and becomes a regular or exceptional fiber in the Seifert fiber space K(r) which may have a degenerate fiber, then c is called a seiferter for (K, r). If K is a nontrivial torus knot, two exceptional fibers in the Seifert fiber space S3 intN(K) are seiferters for any Seifert surgery (K, r). Moreover, if r is an integral slope, then a meridian of K is also a seiferter for (K, r). These seiferters are called basic seiferters for Seifert surgeries on torus knots see Figure 2.1. Surprisingly, we can find seiferters for all known examples of Seifert surgeries except for one infinite family obtained by Teragaito [80] using the Osoinach’s annulus–twisting construction. In this paper, we focus on integral, Seifert surgeries (K, m), where m Z. In Sections 2.1, 2.2 and 2.3, we will prove basic results about seiferters. Twisting a Seifert surgery along its seiferter or an annulus cobounded by a pair of seiferters yields another Seifert surgery, and furthermore the image of the seiferter (resp. the pair of seiferters) remains a seiferter (resp. a pair of seiferters cobounding an annulus) in the resulting surgery (Propositions 2.6, 2.33). Now regard each Seifert surgery as a vertex, and connect two vertices by an edge if they are related by a sin- gle twisting along a seiferter or an annulus cobounded by a pair of seiferters. Then we obtain a 1–dimensional complex, the Seifert Surgery Network. See Section 2.4 for details. 1
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