In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: drilling then filling. Definitions Given a 3-manifold and a link , the manifold drilled along is obtained by removing an open tubular neighborhood of from . If , the drilled manifold has torus boundary components . The manifold drilled along is also known as the link complement, since if one removed the corresponding closed tubular neighborhood from , one obtains a manifold diffeomorphic to . Given a 3-manifold whose boundary is made of 2-tori , we may glue in one solid torus by a homeomorphism (resp. diffeomorphism) of its boundary to each of the torus boundary components 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. In order to describe a Dehn surgery (see ), one picks two oriented simple closed curves and on the corresponding boundary torus of the drilled 3-manifold, where is a meridian of (a curve staying in a small ball in and having linking number +1 with or, equivalently, a curve that bounds a disc that intersects once the component ) and is a longitude of (a curve travelling once along or, equivalently, a curve on such that the algebraic intersection is equal to +1). The curves and generate the fundamental group of the torus , and they form a basis of its first homology group. This gives any simple closed curve on the torus two coordinates and , so that . 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 is called the surgery coefficient of . In the case of links in the 3-sphere or more generally an oriented integral