TG-Hyperbolicity of Composition of Virtual Knots

TL;DR

This work shows that the classical restriction that knot compositions are non-hyperbolic does not extend to virtual knots: the composition of two non-classical tg-hyperbolic virtual knots can be tg-hyperbolic. By introducing corks (singular and nonsingular) and leveraging doubles, incompressible gluing surfaces, and Thurston hyperbolization, the authors prove tg-hyperbolicity for various composition types and obtain explicit lower bounds on hyperbolic volume via vol_{ns} and vol_s. They further show that, using 2-geodesic corks and augmented twist-region constructions, one can realize sequences of tg-hyperbolic compositions with volumes unbounded, particularly for alternating, weakly prime virtual links. The paper provides concrete examples and a framework for estimating or driving volumes of composed virtual knots, highlighting the rich hyperbolic geometry accessible through virtual knot compositions on thickened surfaces.

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.

Figures (25)

• Figure 1: For the corks depicted, the composition with any other knot in a thickened surface cannot be tg-hyperbolic because of the presence of an essential torus.
• Figure 2: An example of nonsingular composition. a) and b) show the diagram and thickened surface representation of the two knots. c) shows each $S_i \times I$ after the cork has been removed, and d) shows the final composition. For simplicity, the inner surface boundary $S_i \times \{0\}$ is not drawn in.
• Figure 3: The virtual trefoil drawn on a torus. A singular curve is drawn in red.
• Figure 4: The result of cutting open a genus one thickened surface containing a singular knot. We cut along the once-punctured annulus $A$ defined by the singular curve $\gamma$ shown in Figure \ref{['fig: virt tref singular curve']}.
• Figure 5: An example of a choice of cork (shown in red) that visually appears to be nonsingular, but is actually a singular cork. The nonsingular composition of the knot with itself has an obvious reducing annulus.

Theorems & Definitions (51)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Definition 2.1
• Definition 2.2
• Lemma 3.1
• proof