Cubes with knotted holes

Abstract

The statement that a knot K has Property P means that (1) if C is a cube with a ^-knotted hole, and M3 is a simply connected 3-manifold obtained by sewing a solid torus onto C, then M3^S3, and (2) any piecewise linear homeomorphism of a cube with a Ä-knotted hole into S3 can be extended to a piecewise linear homeomorphism of S" onto S3.In this paper it is shown that many familiar classes of knots have Property P.1. Introduction.The 3-dimensional Poincaré conjecture is that a compact, connected, simply connected 3-manifold without boundary is topologically a 3sphere S3.Despite efforts to prove the conjecture, it has withstood attack [4].Since no solution has been found, some believe that the conjecture is false, and that there is a counterexample.In attempting to construct a counterexample to the Poincaré conjecture, the first hurdle seems to be that of showing that the proposed example has a trivial fundamental group.This is difficult in that the information about the fundamental group usually appears as an exotic presentation of the group.Having somehow verified that this group is indeed trivial, one encounters the second hurdle of somehow showing that the example is topologically different from S3.It is known that every orientable 3-manifold may be obtained by removing a collection of disjoint solid tori from S3 and sewing them back differently [14], [22], [12].In this paper we examine some of the possibilities for constructing a counterexample to the Poincaré conjecture by removing a single solid torus from S3 and sewing it back differently.Actually, we examine not only this process but one analogous to it which we call "attaching a pillbox to a cube with a knotted hole".We find it useful to consider two sorts of surgery which are inverses of each other.If Ma is a 3-manifold with boundary, A2 is an annulus on Bd M3 and P3 is a 3-cell such that M3 n P3 = A2 = BdP3 r\ Bd M3, then we say that M3 u P3 was obtained by sewing a pillbox P3 to M3 along A2.Suppose that H3 is a 3-cell in M3 such that H3 n Bd M3 is the union of two disjoint disks D\ and D\, and N3 = C\ (M3 -H3) is a 3-manifold with boundary.We say that N3 was obtained by boring a hole in M3.