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 S 3 to knots in S × I where S is a closed orientable surface. In particular, we prove that virtual knots behave very differently under composition. The composition of two hyperbolic virtual knots is hyperbolic. We then obtain strong lower bounds on the volume of the composition using information about the original knots.