Volume 18, issue 4 (2018)

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On hyperbolic knots in $S^3$ with exceptional surgeries at maximal distance

Benjamin Audoux, Ana G Lecuona and Fionntan Roukema

Algebraic & Geometric Topology 18 (2018) 2371–2417

Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5–chain link. We enumerate all hyperbolic knots in S3 obtained by surgery on the minimally twisted 5–chain link that realize the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal) and (lens, Seifert fibred) pairs. In light of Baker’s work, the classification in this paper conjecturally accounts for “most” hyperbolic knots in S3 realizing the maximal distance between these exceptional pairs. As a byproduct, we obtain that all examples that arise from the 5–chain link actually arise from the magic manifold. The classification highlights additional examples not mentioned in Martelli and Petronio’s survey of the exceptional fillings on the magic manifold. Of particular interest is an example of a knot with two lens space surgeries that is not obtained by filling the Berge manifold (ie the exterior of the unique hyperbolic knot in a solid torus with two nontrivial surgeries producing solid tori).

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exceptional surgeries, $5$ chain link, maximal distance, $3$–manifolds, hyperbolic knots
Mathematical Subject Classification 2010
Primary: 57M25, 57M50
Received: 21 June 2017
Revised: 26 December 2017
Accepted: 8 January 2018
Published: 26 April 2018
Benjamin Audoux
Institut de Mathématiques de Marseille, CNRS, Centrale Marseille, UMR 7373
Aix-Marseille Université
Ana G Lecuona
Institut de Mathématiques de Marseille, CNRS, Centrale Marseille, UMR 7373
Aix-Marseille Université
Fionntan Roukema
School of Mathematics and Statistics
University of Sheffield
United Kingdom