The realization problem of essential surfaces in knot exteriors

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.

13 pages, 10 figures