J. Berge, Some knots with surgeries yielding lens spaces, unpublished  Home/Search   Document Not in Database   Summary   Related Articles   Check

....curved, there can be at most 24 Seifert fibered Dehn surgeries on a hyperbolic knot. The first known examples of small Seifert manifolds arising from Dehn surgery on hyperbolic knots were given by [FS80] Berge has a construction 4 which produces families of knots with lens space Dehn surgeries [Ber]. Many of these knots are hyperbolic. It is an open question whether or not the Berge knots include all knots with lens space Dehn surgeries. Berge has explicitly described the knots which arise from his construction. Various examples of small Seifert fibered Dehn surgeries have been given by ....

....Knots In this chapter we will describe a construction of knots in S 3 that have a Dehn surgery that is a Seifert fibered space with base S 2 and three or fewer critical fibers. The construction is a generalization of Berge s construction of knots with lens space Dehn surgeries [Ber]. From the definition of the construction, it is not clear that any nontrivial non Berge examples arise, hence we will describe a simple family of nontrivial examples that arise from the construction. We begin with the definitions of some relevant concepts. We will consider only orientable ....

J. Berge, Some knots with surgeries yielding lens spaces, unpublished

....ordinary torus knots are twisted torus knots, there are certainly some non hyperbolic knots among the twisted torus knots. Moreover, certain non trivial (i.e. s 6= 0 and r 1) twisted torus knots are in fact torus knots. The hyperbolicity of twisted torus knots is considered in [Dea] 3.1. Fibering Twisted Torus Knots. If s 0 then T (p; q) r;s has a description as a closed positive braid as in Figure 1. But this implies that T is a bered knot with ber genus equal to ( p 1) q 1) sr(r 1) 2 [Sta78] If s 0 then a twisted torus knot may or may not be bered. If jsrj q then all ....

....(i.e. s 6= 0 and r 1) twisted torus knots are in fact torus knots. The hyperbolicity of twisted torus knots is considered in [Dea] 3.1. Fibering Twisted Torus Knots. If s 0 then T (p; q) r;s has a description as a closed positive braid as in Figure 1. But this implies that T is a bered knot with ber genus equal to ( p 1) q 1) sr(r 1) 2 [Sta78] If s 0 then a twisted torus knot may or may not be bered. If jsrj q then all negative crossings may be cancelled with positive crossings, and a closed positive braid results. In this case, then, the knot is bered and the same ....

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J. Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript.

....However, there is some evidence that close braids that admit a G flype cannot yield such simple manifolds under Dehn surgery. It is known that surgery on a torus knot can produce a lens space. More generally, surgery on one bridge knots contained in a solid torus can also produce a lens space [B] torus knots are a sub class of such one bridge knots. In [L] it was established that any closed braid representative of a torus knot can be reduced to minimal braid index through the uses of destabilizations and exchange moves, i.e. G flypes are not needed. As applied to torus knots, ....

J. Berge, Some knots with surgeries yielding lens spaces, preprint.

....the Cyclic Surgery Theorem [4] and # 1 K(r) has the order r . Furthermore, there are at most two such slopes, and if there are two then they are consecutive. 2 Theorem 1.1. Let K be a hyperbolic knot in S 3 . If K(r) is a lens space, then r # 12g 7, where g is the genus of K. In [2], Berge introduced double primitive knots and showed that such knots admit integral surgeries which yield lens spaces. Furthermore, he gave a list of double primitive knots, including all known knots with surgeries yielding lens spaces. As he wrote there, there is still a possibility that his ....

J. Berge. Some knots with surgeries yielding lens spaces, unpublished manuscript.

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