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.

Figures (10)

• Figure 1: Positive intersection
• Figure 2: Diagram $\mathcal{D}$
• Figure 3: Parallel copies of the slope $1/2$ over the regular neighborhood of the tangle $1/2$.
• Figure 4: Example of a saddle move from slope $1/7$ to slope $0$.
• Figure 5: Taking parallel copies of $\alpha$ and numbering from leftmost to rigthmost

Theorems & Definitions (16)

• Conjecture 1.2: Strong Cabling Conjecture
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Definition 3.1
• Theorem 3.2
• proof
• Corollary 3.3: Corollary 2.5 from HO