In the drilling process one removes an open tubular neighbourhood of the link from the 3-manifold. The resulting manifold we call the link complement.
Given a 3-manifold with torus boundary components, 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 process is called Dehn filling.
Dehn surgery on a 3-manifold containing a link consists of drilling out a tubular neighbourhood of the link together with Dehn filling on all the components of the boundary corresponding to the link.
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 . As long as the meridian maps to the surgery slope , 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 links in the 3-sphere or more generally an oriented homology sphere, there is a canonical choice of the meridians and longitudes of given by a Seifert surface. When the ratio p/q are all integers, the surgery is called an integral surgery or a genuine surgery, since such surgeries are closely related to handlebodies and cobordism.