Dehn surgery
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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 T of the original 3-manifold. There are many inequivalent ways of doing this, in general. This is called a Dehn surgery or Dehn filling, although the latter mostly occurs in contexts related to hyperbolic geometry.
We can pick two oriented simple closed curves m and l on the boundary torus of the 3-manifold that generate the fundamental group of the torus. This gives any simple closed curve γ on that torus two coordinates p and q, each coordinate corresponding to the algebraic intersection of the curve with m and l respectively. These coordinates only depend on the homotopy class of γ.
We can specify a homeomorphism of the boundary of a solid torus to T by having the meridian curve of the solid torus map to a curve homotopic to γ. As long as the meridian maps to the surgery slope [γ], 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 m to be the meridian of a solid torus neighborhood of a link component and l to be the longitude.
