Boundary-incompressible surface
Encyclopedia
Suppose S is a compact surface with boundary that is properly embedded in a 3-manifold M with boundary. Suppose further that there exists a disk D in M such that 
and 
are arcs in 
, with 
, 
, and 
is an essential arc in S (
does not cobound a disk in S with another arc in 
). Then D is called a boundary-compressing disk for S in M.
The surface S is said to be boundary-compressible if either S is a disk that cobounds a ball with a disk in
or there exists a boundary-compresssing 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
). 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
or there exists a boundary-compresssing 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
, 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 
and there does not exist a boundary-compressing disk for S in V, so S is boundary-incompressible by the second definition.
and are arcs in , with , , and is an essential arc in S ( does not cobound a disk in S with another arc in ). Then D is called a boundary-compressing disk for S in M.The surface S is said to be boundary-compressible if either S is a disk that cobounds a ball with a disk in
or there exists a boundary-compresssing 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
). 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
or there exists a boundary-compresssing disk for S in M. Otherwise, S is boundary-incompressible.For instance, if K is a trefoil knot
Trefoil knot
In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...
embedded in the boundary of a solid torus V and S is the closure of a small annular neighborhood of K in
, 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 and there does not exist a boundary-compressing disk for S in V, so S is boundary-incompressible by the second definition.