On multiply twisted knots that are Seifert fibered or toroidal

TL;DR

This paper classifies certain highly twisted knots based on their geometric structures, identifying when they are Seifert fibered or toroidal, and analyzes their decompositions and Gromov norms.

Contribution

It provides a classification of generalized augmented links that are Seifert fibered or toroidal, extending understanding of highly twisted knots and their geometric properties.

Findings

Classified Seifert fibered generalized augmented links.

Established torus decompositions for toroidal cases.

Derived lower bounds on Gromov norms of knot complements.

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.