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 ).
|Date of creation||2013-03-22 13:56:07|
|Last modified on||2013-03-22 13:56:07|
|Last modified by||bwebste (988)|