Knot exteriors with all compact surfaces of positive genus essentially embedded

TL;DR

This paper demonstrates the existence of infinitely many knot exteriors that contain essential surfaces of any positive genus and boundary components, expanding understanding of surface embeddings in knot complements.

Contribution

It introduces new examples of knot exteriors with all compact surfaces of positive genus essentially embedded, highlighting their diverse topological properties.

Findings

Existence of infinitely many such knot exteriors.

Presence of longitudinal essential surfaces of arbitrary positive genus.

Surfaces with any number of boundary components are embedded.

Abstract

It is well known that there exist knots with Seifert surfaces of arbitrarily high genus. In this paper, we show the existence of infinitely many knot exteriors where each of which has longitudinal essential surfaces of any positive genus and any number of boundary components.