Densities of Hyperbolic Cusp Invariants

Colin Adams; Rose Kaplan-Kelly; Michael Moore; Brandon Shapiro; Shruthi Sridhar; Joshua Wakefield

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arXiv · 1701.03479

Densities of Hyperbolic Cusp Invariants

Colin Adams·Rose Kaplan-Kelly·Michael Moore·Brandon Shapiro·Shruthi Sridhar·Joshua Wakefield

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Abstract

We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing density for links is bounded above by 3.1263.... Moreover, there is a sequence of links with cusp crossing density approaching 3. The least upper bound for cusp crossing density remains an open question. For two-component hyperbolic links, cusp crossing density is shown to be dense in the interval [0,1.6923...] and for all hyperbolic links, cusp crossing density is shown to be dense in [0, 2.120...].

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Colin Adams, Rose Kaplan-Kelly, Michael Moore, Brandon Shapiro, Shruthi Sridhar, Joshua Wakefield. 2017-01-12. Densities of Hyperbolic Cusp Invariants. https://arxiv.org/abs/1701.03479

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