TG-Hyperbolicity of Composition of Virtual Knots
Abstract
The composition of any two nontrivial classical knots is a satellite knot, and thus, by work of Thurston, is not hyperbolic. In this paper, we explore the composition of virtual knots, which are an extension of classical knots that generalize the idea of knots in to knots in where is a closed orientable surface. We prove that for any two hyperbolic virtual knots, there is a composition that is hyperbolic. We then obtain strong lower bounds on the volume of the composition using information from the original knots.
Cite
@article{arxiv.2110.09859,
title = {TG-Hyperbolicity of Composition of Virtual Knots},
author = {Colin Adams and Alexander Simons},
journal= {arXiv preprint arXiv:2110.09859},
year = {2025}
}
Comments
33 pages, 25 figures This new revision is substantially different from the initial version. In particular, Theorem 3.7 and Lemma 3.1 of that initial version are not correct as stated. The revision corrects those results and proves additional results