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Networking Seifert Surgeries on Knots
About this Title
Arnaud Deruelle, Institute of Natural Sciences, Nihon University, Tokyo 156–8550, Japan, Katura Miyazaki, Faculty of Engineering, Tokyo Denki University, Tokyo 101–8457, Japan and Kimihiko Motegi, Department of Mathematics, Nihon University, Tokyo 156–8550, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 217, Number 1021
ISBNs: 978-0-8218-5333-7 (print); 978-0-8218-8754-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00635-0
Published electronically: September 12, 2011
Keywords: Dehn surgery,
hyperbolic knot,
Seifert fiber space,
seiferter,
Seifert Surgery Network
MSC: Primary 57M25, 57M50; Secondary 57N10
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction
- 2. Seiferters and Seifert Surgery Network
- 3. Classification of seiferters
- 4. Geometric aspects of seiferters
- 5. $S$–linear trees
- 6. Combinatorial structure of Seifert Surgery Network
- 7. Asymmetric seiferters and Seifert surgeries on knots without symmetry
- 8. Seifert surgeries on torus knots and graph knots
- 9. Paths from various known Seifert surgeries to those on torus knots
Abstract
We propose a new approach in studying Dehn surgeries on knots in the $3$–sphere $S^3$ 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 $1$–dimensional complex whose vertices correspond to Seifert surgeries. A seiferter for a Seifert surgery on a knot $K$ is a trivial knot in $S^3$ disjoint from $K$ that becomes a fiber in the resulting Seifert fiber space. Twisting $K$ 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 $S^3$.
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