Surface knot

A surface knot is a smooth embedding of a closed connected surface in a four-manifold. In the narrow standard usage, a 2-knot is an embedded two-sphere in $S^4$.

It extends the embedding viewpoint of knot theory to two-dimensional domains.

A banded-unlink diagram consists of an unlink $U$ and disjoint bands whose simultaneous surgery produces another unlink. Capping the lower and upper unlinks by disks and tracing the band surgeries constructs a surface in $S^4$.