Unexpected Essential Surfaces among Exteriors of Twisted Torus Knots

TL;DR

This paper discovers new families of twisted torus knots with essential tori in their exteriors, showing they do not all belong to previously known categories, thus expanding understanding of their topology.

Contribution

It introduces two novel families of toroidal twisted torus knots, providing counterexamples to Morimoto's classification question.

Findings

Identified two new families of toroidal twisted torus knots

Demonstrated these families are not part of Morimoto's two known classes

Expanded the classification of twisted torus knots with essential tori

Abstract

The twisted torus knots K(p, q; r, s) are obtained by performing a sequence of s full twists on r adjacent strands of (p, q)-torus knots. Morimoto asked whether all twisted torus knots with essential tori in the exterior fit into one of two families. We prove that the answer to this question is no, by finding two different new families of toroidal twisted torus knots.