On the maximal number and the diameter of exceptional surgery slope sets

TL;DR

This paper explores the relationship between the maximum size and diameter of exceptional surgery slope sets for hyperbolic knots, showing that the diameter bound implies the maximum cardinality bound and identifying a special slope related to shortest geodesics.

Contribution

It demonstrates that the diameter bound of 8 implies the maximal cardinality bound of 10 for exceptional surgery slopes, and identifies a slope close to all exceptional slopes in generic cases.

Findings

Diameter bound of 8 implies maximum cardinality of 10.

Existence of a slope with all exceptional slopes within distance two.

In generic cases, this slope can be the shortest geodesic on a horotorus.

Abstract

Concerning the set of exceptional surgery slopes for a hyperbolic knot, Lackenby and Meyerhoff proved that the maximal cardinality is 10 and the maximal diameter is 8. Their proof is computer-aided in part, and both bounds are achieved simultaneously. In this note, it is observed that the diameter bound 8 implies the maximal cardinality bound 10 for exceptional surgery slope sets. This follows from the next known fact: For a hyperbolic knot, there exists a slope on the peripheral torus such that all exceptional surgery slopes have distance at most two from the slope. We also show that, in generic cases, that particular slope above can be taken as the slope represented by the shortest geodesic on a horotorus in a hyperbolic knot complement.