TG-Hyperbolicity of Composition of Virtual Knots

TL;DR

This paper investigates the hyperbolic properties of composed virtual knots, demonstrating that the composition of two hyperbolic virtual knots can also be hyperbolic and establishing volume bounds based on original knots.

Contribution

It proves that the composition of two hyperbolic virtual knots can be hyperbolic, extending classical knot results to virtual knots, and provides volume bounds for such compositions.

Findings

Composition of two hyperbolic virtual knots can be hyperbolic.

Established lower bounds on the volume of composed virtual knots.

Extended classical hyperbolic knot results to 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 $S^3$ to knots in $S \times I$ where $S$ 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.