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Networking Seifert Surgeries on Knots

Arnaud Deruelle Nihon University, Tokyo, Japan
Katura Miyazaki Tokyo Denki University, Tokyo, Japan
Kimihiko Motegi Nihon University, Tokyo, Japan
Available Formats:
Electronic ISBN: 978-0-8218-8754-7
Product Code: MEMO/217/1021.E
List Price: $71.00 MAA Member Price:$63.90
AMS Member Price: $42.60 Click above image for expanded view Networking Seifert Surgeries on Knots Arnaud Deruelle Nihon University, Tokyo, Japan Katura Miyazaki Tokyo Denki University, Tokyo, Japan Kimihiko Motegi Nihon University, Tokyo, Japan Available Formats:  Electronic ISBN: 978-0-8218-8754-7 Product Code: MEMO/217/1021.E  List Price:$71.00 MAA Member Price: $63.90 AMS Member Price:$42.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2172012; 130 pp
MSC: Primary 57;

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 $S^3$.

• 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
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Volume: 2172012; 130 pp
MSC: Primary 57;

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 $S^3$.

• 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
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