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ISSN 1947-6221(e) ISSN 0065-9266(p)

Networking Seifert Surgeries on Knots

Authors: Arnaud Deruelle, Katura Miyazaki and Kimihiko Motegi
Journal: Memoirs of the AMS 217 (2012)
MSC (2010): Primary 57M25, 57M50; Secondary 57N10
Posted: September 12, 2011
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Abstract: We propose a new approach in studying Dehn surgeries on knots in the -sphere yielding Seifert fiber spaces. Our basic idea is finding relationships among such surgeries. To describe relationships and get a global picture of Seifert surgeries, we introduce ``seiferters'' and the Seifert Surgery Network, a -dimensional complex whose vertices correspond to Seifert surgeries. A seiferter for a Seifert surgery on a knot is a trivial knot in disjoint from that becomes a fiber in the resulting Seifert fiber space. Twisting along its seiferter or an annulus cobounded by a pair of its seiferters yields another knot admitting a Seifert surgery. Edges of the network correspond to such twistings. A path in the network from one Seifert surgery to another explains how the former Seifert surgery is obtained from the latter after a sequence of twistings along seiferters and/or annuli cobounded by pairs of seiferters. We find explicit paths from various known Seifert surgeries to those on torus knots, the most basic Seifert surgeries.

We classify seiferters and obtain some fundamental results on the structure of the Seifert Surgery Network. From the networking viewpoint, we find an infinite family of Seifert surgeries on hyperbolic knots which cannot be embedded in a genus two Heegaard surface of .

References

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