Networking Seifert surgeries on knots. II: The Berge's lens surgeries
In an unpublished manuscript ``Networking Seifert surgeries on knots'' (available on the third author's webpage), the authors defined a 1-dimensional complex whose vertices are Seifert surgeries; that is, Dehn surgeries on a knot in the 3-sphere resulting in Seifert fiber spaces. A Seifert for a Seifert surgery is a trivial knot in the 3-sphere that becomes a fiber in the resulting Seifert fiber space. Two vertices of the complex are connected by an edge if one is obtained from the other by twisting along a Seifert. The complex is called the Seifert surgery network. The present paper is concerned with surgeries that yield lens spaces. J. Berge, in an unpublished manuscript from 1990, gave a list of twelve infinite families of such surgeries and it has been conjectured that the list is complete [cf. \textit{C. McA. Gordon}, Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 631--642 (1991; Zbl 0743.57008)]. The authors show that any surgery on Berge's list can be obtained from a Seifert surgery on a torus knot by twisting along at most two Seiferts.
scientific article; zbMATH DE number 5539530
scientific article