Unexpected essential surfaces among exteriors of twisted torus knots

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. In this paper, we answer two questions related to essential surfaces in the exteriors of twisted torus knots. Namely, we show there are prime numbers r greater than 2 such that K(p, q; r, s) contain closed essential surface in their exterior, answering a question of Morimoto and Yamada. Additionally, Morimoto asked whether all twisted torus knots with essential tori in the exterior fit into one of two families. We find a new infinite family that was previously unknown.