Boundary-incompressible surface
In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot be made simpler by a certain type of operation known as boundary compression. Suppose "M" is a 3-manifold with boundary. Suppose also that "S" is a compact surface with boundary that is properly embedded in "M", meaning that the boundary of "S" is a subset of the boundary of "M" and the interior points of "S" are a subset of the interior points of "M". A boundary-compressing disk for "S" in "M" is  defined to be a disk "D" in "M" such that formula_1 and formula_2 are arcs in formula_3, with formula_4, formula_5, and formula_6 is an essential arc in "S" (formula_6 does not cobound a disk in "S" with another arc in formula_8). The surface "S" is said to be boundary-compressible if either "S" is a disk that cobounds a ball with a disk in formula_9 or there exists a boundary-compressing disk for "S" in "M". Otherwise, "S" is boundary-incompressible. Alternatively, one can relax this definition by dropping the requirement that the surface be properly embedded. Suppose now that  S is a compact surface (with boundary) embedded in the boundary of a 3-manifold "M". Suppose further that "D" is a properly embedded disk in "M" such that "D" intersects "S" in an essential arc (one that does not cobound a disk in "S" with another arc in formula_8). Then "D" is called a boundary-compressing disk for "S" in "M". As above, "S" is said to be boundary-compressible if either "S" is a disk in formula_9 or there exists a boundary-compressing disk for "S" in "M". Otherwise, "S" is boundary-incompressible. For instance, if "K" is  a trefoil knot embedded in the boundary of a solid torus "V" and "S" is the closure of a small annular neighborhood of "K" in formula_12, then "S" is not properly embedded in "V" since the interior of "S" is not contained in the interior of "V". However, "S" is embedded in formula_12 and there does not exist a boundary-compressing disk for "S" in "V", so "S" is boundary-incompressible by the second definition.