Let be a smooth 3-manifold, and a smooth knot. Since is an embedded submanifold, by the tubular neighborhood theorem there is a closed neighborhood of diffeomorphic to the solid torus . We let denote the interior of . Now, let be an automorphism of the torus, and consider the manifold , which is the disjoint union of and , with points in the boundary of identified with their images in the boundary of under .

It's a bit hard to visualize how this actually results in a different manifold, but it generally does. For example, if , the 3-sphere, is the trivial knot, and is the automorphism exchanging meridians and parallels (i.e., since , get an isomorphism , and is the map interchanging to the two copies of ), then one can check that ( is also a solid torus, and after our automorphism, we glue the two solid tori, meridians to meridians, parallels to parallels, so the two copies of paste along the edges to make ).

Every compact 3-manifold can obtained from the by surgery around finitely many knots.