The signature and cusp geometry of hyperbolic knots
Abstract
We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. We show that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We also show a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times.
Keywords
Cite
@article{arxiv.2111.15323,
title = {The signature and cusp geometry of hyperbolic knots},
author = {Alex Davies and András Juhász and Marc Lackenby and Nenad Tomasev},
journal= {arXiv preprint arXiv:2111.15323},
year = {2024}
}
Comments
28 pages, 13 figures. v3: revised final version. Accepted by Geometry & Topology