# Networking Seifert Surgeries on Knots

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*Arnaud Deruelle; Katura Miyazaki; Kimihiko Motegi*

The authors propose a new approach in studying Dehn surgeries on knots in the
\(3\)–sphere \(S^3\) yielding Seifert fiber spaces. The
basic idea is finding relationships among such surgeries. To describe
relationships and get a global picture of Seifert surgeries, they 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. The
authors
find explicit paths from various known Seifert surgeries to those on torus
knots, the most basic Seifert surgeries.

The authors classify seiferters and obtain some fundamental results on the structure
of the Seifert Surgery Network. From the networking viewpoint, they find an
infinite family of Seifert surgeries on hyperbolic knots which cannot be
embedded in a genus two Heegaard surface of \(S^3\).

#### Table of Contents

# Table of Contents

## Networking Seifert Surgeries on Knots

- Acknowledgments vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Seiferters and Seifert Surgery Network 514 free
- Chapter 3. Classification of seiferters 3140
- Chapter 4. Geometric aspects of seiferters 5160
- Chapter 5. S–linear trees 5564
- Chapter 6. Combinatorial structure of Seifert Surgery Network 6776
- Chapter 7. Asymmetric seiferters and Seifert surgeries on knots without symmetry 8392
- Chapter 8. Seifert surgeries on torus knots and graph knots 93102
- Chapter 9. Paths from various known Seifert surgeries to those on torus knots 99108
- 9.1. Lens surgeries given by primitive/primitive construction 99108
- 9.2. Seifert surgeries given by primitive/Seifert–fibered construction 101110
- 9.3. Seifert surgeries given by the Montesinos trick 116125
- 9.4. Toroidal Seifert surgeries over S2 117126
- 9.5. Toroidal Seifert surgeries over RP2 119128

- Bibliography 127136