Dehn surgery

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 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 and on the boundary torus of the 3-manifold that generate the fundamental group of the torus. This gives any simple closed curve on that torus two coordinates p and q, each coordinate corresponding to the algebraic intersection of the curve with and respectively. These coordinates only depend on the homotopy class of .

We can specify a homeomorphism of the boundary of a solid torus to by having the meridian curve of the solid torus map to a curve homotopic to . The resulting Dehn surgery gives a 3-manifold that does not depend, up to homeomorphism, on the specific gluing, as long as the meridian maps to , the surgery slope. p/q is called the surgery coefficient.

In the case of a link complement, it is usual to pick to be the meridian of a solid torus neighborhood of a link component and to be the longitude.

See also