Compressible Surface

Low Dimensional Topology

Let $S$ be a compact surface properly embedded in a smooth 3 - manifold $M$.

Compressible Disk

A compressing disk $D$ is a disk embedded in $M$ such that $D\cap S=\partial D$ and the intersection is transverse.

If the curve $\partial D$ does not bound a disk inside of $S$, then $D$ is called a nontrivial compressing disk.

If $S$ has a nontrivial compressing disk, then we call $S$ a compressible surface in $M$.

If $S$ is neither the $S^2$ nor a compressible surface, then we call the surface incompressible.

Example in stack

$D$ is a compressible disk.

$\partial$-incompressible

A properly embedded surface $S\subset M$ is $\partial$-incompressible if for each disk $D\subset M$ such that $\partial D\cap S$ is an arc $\alpha$ in $\partial D$ and $\partial D\setminus \alpha \subset \partial M$ (such a $D$ is called a $\partial$ - compressing disk for $S$) there is a disk $D^{\prime }\subset S$ with $\alpha \subset \partial D^{\prime }$ and $\partial D^{\prime }\setminus \alpha \subset \partial M$. boundary compressible disk

Essential Surface

A surface which is either incompressible and $\partial$ - incompressible, or a sphere not bounding a ball, or a disk that is not boundary parallel, will be called an essential surface.

Given a closed surface $F$ in $S^3$ containing a link $L$, we say that $F$ is essential with respect to $L$ when $F\cap \left(S^3\setminus \nu \left(L\right)\right)$ is essential in $S^3\setminus \nu \left(L\right)$. When $L$ is clear, we may simply write that $F$ is essential.