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Networking Seifert surgeries on knots, III

Arnaud Deruelle, Katura Miyazaki and Kimihiko Motegi

Algebraic & Geometric Topology 14 (2014) 2065–2101

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint introduced by the authors in [Mem. Amer. Math. Soc. 217 (2012), no. 1021]. The Seifert surgery network is a 1–dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a “hyperbolic seiferter” or a “hyperbolic annular pair” produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots that have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted.

Dehn surgery, hyperbolic knot, Seifert fiber space, seiferter, Seifert surgery network, band-sum
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M50, 57N10
Received: 8 November 2012
Revised: 8 November 2013
Accepted: 17 November 2013
Published: 28 August 2014
Arnaud Deruelle
Institute of Natural Sciences
Nihon University
3-25-40 Sakurajosui, Setagaya-ku
Tokyo 156-8550
Katura Miyazaki
Faculty of Engineering
Tokyo Denki University
5 Senju-Asahi-cho, Adachi-ku
Tokyo 120-8551
Kimihiko Motegi
Department of Mathematics
Nihon University
3-25-40 Sakurajosui, Setagaya-ku
Tokyo 156-8550