Knot exteriors with all possible meridional essential surfaces

TL;DR

This paper demonstrates the existence of infinitely many knot exteriors containing meridional essential surfaces of arbitrary genus and boundary components, including a hyperbolic knot exterior with unbounded surface complexity.

Contribution

It establishes the existence of knot exteriors with all possible meridional essential surfaces, extending understanding of surface embeddings in knot complements.

Findings

Existence of infinitely many knot exteriors with all meridional essential surfaces.

Construction of a hyperbolic knot exterior with unbounded genus and boundary components.

Meridional essential surfaces embed into these knot exteriors regardless of their complexity.

Abstract

We show the existence of infinitely many knot exteriors where each of which contains meridional essential surfaces of any genus and (even) number of boundary components. That is, the compact surfaces that have a meridional essential embedding into a knot exterior have a meridional essential embedding into each of these knot exteriors. From these results, we also prove the existence of a hyperbolic knot exterior in some 3-manifold for which there are meridional essential surfaces of independently unbounded genus and number of boundary components.