On the maximal number and the diameter of exceptional surgery slope sets
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.
Keywords
Cite
@article{arxiv.1110.0572,
title = {On the maximal number and the diameter of exceptional surgery slope sets},
author = {Kazuhiro Ichihara},
journal= {arXiv preprint arXiv:1110.0572},
year = {2012}
}
Comments
5 pages. v2: the proposition in section 3 is replaced, since the proof of that is v1 contains a gap. v3: minor changes. To appear in Proceeding of the Institute of Natural Sciences, Nihon University