An incompressible surface is a properly embedded two-sided topological surface with no compressing disk, subject to the usual exclusion of inessential spherical components. A compressing disk has boundary an essential closed curve on the topological surface and interior disjoint from it.
In an orientable irreducible Seifert fibered space, an essential two-sided incompressible surface that is also a boundary-incompressible surface can be isotoped to a horizontal surface in a Seifert fibered space or a vertical surface in a Seifert fibered space.
A properly embedded topological surface is boundary-incompressible when no disk in the ambient three-manifold joins an essential arc on the topological surface to an arc on the ambient boundary, with the disk's interior disjoint from both.
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An **incompressible surface** is a concept from the field of topology, specifically in the study of 3-manifolds. It refers to a two-dimensional surface that cannot be compressed into a simpler form without cutting it. This property is significant in both mathematical theory and applications, such as in knot theory and the study of 3-manifolds.