Seifert surgeries on knots and their network

Kimihiko Motegi

Knots in Washington XXVII; 3rd Japan-USA Workshop in Knot Theory
January 9-11, 2009
George Washington University
Washington, DC, USA

Organizers
Yoshiyuki Ohyama (TWCU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Alexander Shumakovitch (GWU), Kouki Taniyama (Waseda and GWU), Tatsuya Tsukamoto (Osaka IT), Hao Wu (GWU), Akira Yasuhara (Tokyo GU)

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Seifert surgeries on knots and their network
by
Kimihiko Motegi
Nihon University, Japan
Coauthors: Arnaud Deruelle, Katura Miyazaki

Let K be a hyperbolic knot in the 3-sphere having a Seifert surgery, i.e. a Dehn surgery yielding a Seifert fiber space. There are several ways to show that the resulting manifold is actually Seifert fibered. On the other hand, there was no way to explain "why" the hyperbolic knot K has such a Seifert surgery. To find a natural explanation to the production of Seifert surgeries, we have introduced the Seifert Surgery Network in which a vertex is a pair (K, m) of a knot K and an integer m such that the result of the m-surgery on K is a Seifert fiber space, where a Seifert fiber space may have a fiber of index zero as a degenerate fiber. The networking viewpoint enables us to draw a global picture of Seifert surgeries and clarify relationships among those Seifert surgeries. In particular, if we have a path from a Seifert surgery (K, m) on a hyperbolic knot K to a Seifert surgery (T_{p, q}, n) on a torus knot T_{p, q}, then we can regard (T_{p, q}, n) as an "origin" of (K, m) and the path explains the production of (K, m). In this talk, we look at some particular examples and then present a list of Seifert surgeries on torus knots which are "origins" of those on hyperbolic knots. We will also discuss some related results.

Date received: October 31, 2008