• Given a
3-manifold M and a
link L \subset M, the manifold
M drilled along L is obtained by removing an open
tubular neighborhood of L from M. If L = L_1\cup\dots\cup L_k , the drilled manifold has k torus boundary components T_1\cup\dots\cup T_k. The manifold
M drilled along L is also known as the
link complement, since if one removed the corresponding closed tubular neighborhood from M, one obtains a manifold diffeomorphic to M \setminus L. • Given a 3-manifold whose boundary is made of 2-tori T_1\cup\dots\cup T_k, we may glue in one
solid torus by a
homeomorphism (resp.
diffeomorphism) of its boundary to each of the torus boundary components T_i 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, one picks two oriented simple closed
curves m_i and \ell_i on the corresponding boundary torus T_i of the drilled 3-manifold, where m_i is a meridian of L_i (a curve staying in a small ball in M and having linking number +1 with L_i or, equivalently, a curve that bounds a disc that intersects once the component L_i) and \ell_i is a longitude of T_i (a curve travelling once along L_i or, equivalently, a curve on T_i such that the algebraic intersection \langle\ell_i, m_i\rangle is equal to +1). The curves m_i and \ell_i generate the
fundamental group of the torus T_i, and they form a basis of its first
homology group. This gives any simple closed curve \gamma_i on the torus T_i two coordinates a_i and b_i, so that [\gamma_i] = [a_i \ell_i+b_i m_i]. These coordinates only depend on the
homotopy class of \gamma_i. We can specify a homeomorphism of the boundary of a solid torus to T_i by having the meridian curve of the solid torus map to a curve homotopic to \gamma_i. As long as the meridian maps to the
surgery slope [\gamma_i], the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio b_i/a_i\in\mathbb{Q}\cup\{\infty\} is called the
surgery coefficient of L_i. In the case of links in the 3-sphere or more generally an oriented integral homology sphere, there is a canonical choice of the longitudes \ell_i: every longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a
Seifert surface. When the ratios b_i/a_i are all integers (note that this condition does not depend on the choice of the longitudes, since it corresponds to the new meridians intersecting exactly once the ancient meridians), the surgery is called an
integral surgery. Such surgeries are closely related to
handlebodies,
cobordism and
Morse functions. ==Examples==