Infinitely many meridional essential surfaces of bounded genus in hyperbolic knot exteriors

TL;DR

This paper demonstrates the existence of infinitely many hyperbolic knots with a rich variety of meridional essential surfaces of arbitrary genus and boundary components, revealing complex topological structures in their exteriors.

Contribution

It constructs examples of hyperbolic knots with infinitely many meridional essential surfaces of any genus and boundary component count, expanding understanding of knot exteriors.

Findings

Existence of infinitely many hyperbolic knots with complex essential surfaces

Presence of essential tangle decompositions with arbitrarily many strings

Surfaces of any positive genus and boundary components embedded meridionally

Abstract

We show the existence of an infinite collection of hyperbolic knots where each of which has in its exterior meridional essential planar surfaces of arbitrarily large number of boundary components, or, equivalently, that each of these knots has essential tangle decompositions of arbitrarily large number of strings. Moreover, each of these knots has in its exterior meridional essential surfaces of any positive genus and (even) number of boundary components. That is, the compact surfaces that have a meridional essential embedding into a hyperbolic knot exterior have meridional essential embeddings into each of these hyperbolic knots exteriors.