On multiply twisted knots that are Seifert fibered or toroidal

Abstract

Abstract. We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either "trivial", in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high Dehn filling, giving results on the torus decomposition of knots with generalized twist regions and a high amount of twisting. As an application, we give lower bounds on the Gromov norms of these knot complements and of generalized augmented links.

An introduction to fully augmented links

“…The existence of this surface and the reflection allows us to obtain some information on volumes and geometry of these links. These are explored in [11] and in [14].…”

Hyperbolic geometry of multiply twisted knots

“…Finally, note that the focus of this paper is on geometric information on hyperbolic generalized augmented links and their Dehn fillings. In a companion paper, we discuss results for generalized augmented links which are not hyperbolic [20].…”