Dehn surgery
A Dehn surgery is a specific construction used to modify 3-manifolds with at least one torus boundary component, e.g. link complements.
Since there is a torus boundary component, we may glue in a solid torus by a homeomorphism of its boundary to the torus boundary component <math>T<math> of the original 3-manifold. There are many inequivalent ways of doing this, in general. This is called a Dehn surgery or Dehn filling.
We can pick two oriented simple closed curves <math>m<math> and <math>l<math> on the boundary torus of the 3-manifold that generate the fundamental group of the torus. This gives any simple closed curve <math>\gamma<math> on that torus two coordinates p and q, each coordinate corresponding to the algebraic intersection of the curve with <math>m<math> and <math>l<math> respectively. These coordinates only depend on the homotopy class of <math>\gamma<math>.
We can specify a homeomorphism of the boundary of a solid torus to <math>T<math> by having the meridian curve of the solid torus map to a curve homotopic to <math>\gamma<math>. As long as the meridian maps to the surgery slope <math>[\gamma]<math>, the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio p/q is called the surgery coefficient.
In the case of a link complement, it is usual to pick <math>m<math> to be the meridian of a solid torus neighborhood of a link component and <math>l<math> to be the longitude.
See also
Categories: 3-manifolds | Mathematics stubs