Essential meridional surfaces for tunnel number one knots

TL;DR

This paper demonstrates the existence of infinitely many tunnel number one knots with essential meridional surfaces of specified genus and boundary components, and also constructs knots with multiple disjoint incompressible surfaces of the same genus.

Contribution

It establishes new existence results for essential meridional surfaces and disjoint incompressible surfaces within tunnel number one knot exteriors.

Findings

Infinitely many tunnel number one knots with essential meridional surfaces of genus g and 2n boundary components.

Existence of tunnel number one knots with n disjoint, non-parallel, incompressible surfaces of genus n.

Construction methods for such knots and surfaces.

Abstract

We show that for each pair of positive integers g and n, there are infinitely many tunnel number one knots, whose exteriors contain an essential meridional surface of genus g, and with 2n boundary components. We also show that for each positive integer n, there are tunnel number one knots whose exteriors contain n disjoint, non-parallel, closed incompressible surfaces, each of genus n.