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

Abstract

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 $S^3$ 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 $S^3$ 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).