The realization problem of essential surfaces in knot exteriors

TL;DR

This paper investigates which combinations of genus, boundary components, and boundary slope can be realized by essential surfaces in knot exteriors, establishing existence results for even boundary components and boundary slopes.

Contribution

It proves that for any even number of boundary components and any boundary slope, there exists a knot with an essential surface matching these parameters.

Findings

If boundary components are odd, boundary slope must be 1.

Existence of essential surfaces for any even boundary components and boundary slope.

Not all parameter combinations are realizable, with specific restrictions.

Abstract

We study compact orientable essential surfaces in knot exteriors in the 3-sphere. The genus $g$, the number of boundary components $b$, and the boundary slope $p/q$ are fundamental invariants of an essential surface. The \textit{realization problem} asks whether, for a given triple $(g, b, q)$ with $g \ge 0$, $b \ge 1$, and $q \ge 1$, there exists a knot $K \subset S^3$ whose exterior $E(K)$ contains a compact orientable essential surface $F$ of genus $g$ with $b$ boundary components and boundary slope $p/q$ for some $p$. In general, not all combinations of $(g, b, q)$ are realizable. First, we show that if $b$ is odd, then $q$ must be equal to $1$. Our main theorem states that for any given even $b \ge 2$ and $q \ge 1$, there exist a genus $g \ge 0$ and a knot $K$ such that $E(K)$ contains a compact orientable essential surface with these parameters.