In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be simplified. In non-mathematical terms, the surface of a suitcase is compressible, because we could cut the handle and shrink it into the surface. But a Conway sphere (a sphere with four holes) is incompressible, because there are essential parts of a knot or link both inside and out, so there is no way to move the entire knot or link to one side of the punctured sphere. The mathematical definition is as follows. There are two cases to consider. A sphere is incompressible if both inside and outside the sphere there are some obstructions that prevent the sphere from shrinking to a point and also prevent the sphere from expanding to encompass all of space. A surface other than a sphere is incompressible if any disk with its boundary on the surface spans a disk in the surface. Incompressible surfaces are used for decomposition of Haken manifolds, in normal surface theory, and in the study of the fundamental groups of 3-manifolds. Formal definition Let S be a compact surface properly embedded in a smooth or PL 3-manifold M. A compressing disk D is a disk embedded in M such that and the intersection is transverse. If the curve ∂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 2-sphere nor a compressible surface, then we call the surface (geometrically) incompressible. Note that 2-spheres are excluded since they have no nontrivial compressing disks by the Jordan-Schoenflies theorem, and 3-manifolds have abundant embedded 2-spheres. Sometimes one alters the definition so that an incompressible sphere is a 2-sphere embedded in a 3-manifold that does not bound an embedded 3-ball. Such spheres arise exactly when a 3-manifold is not irreducible. Since this notion of incompressibility for a sphere is quite different from the above definition for surfaces, often an incompressible sphere is instead referred to as an essential sphere or a reducing sphere. Compression Given a compressible surface S with a compressing disk D that we may assume lies in the interior of M and intersects S transversely, one may perform embedded 1-surgery on S to get a surface that is obtained