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

## TL;DR

This paper classifies hyperbolic knots in S^3 obtained via surgery on the minimally twisted 5-chain link that realize maximal distances between certain exceptional slope pairs, expanding understanding of such knots beyond known Berge knots.

## Contribution

It enumerates all hyperbolic knots from the minimally twisted 5-chain link with maximal slope distance, including new examples not derived from Berge or the Berge manifold.

## Key findings

- All examples are realized by filling the magic manifold.
- Identification of additional exceptional fillings on the magic manifold.
- Discovery of a knot with two lens space surgeries not from the Berge manifold.

## Abstract

Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article 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), (lens, Seifert fibred spaces) 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. All examples obtained in our classification are realized by filling 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.