A simply connected 3-manifold is 𝑆³ if it is the sum of a solid torus and the complement of a torus knot

Abstract

It has been shown [6] that any closed, connected, orientable 3manifold can be constructed by removing a finite number, k, of mutually exclusive tame solid tori1 from the 3-sphere, S3, and then sewing them back in some possibly different manner. In particular, any homotopy 3-sphere can be obtained in this manner; thus it gives an approach to the Poincar6 Conjecture. We wish to consider the case k =1. To this end let T be a tame solid torus in SI and let M be a simply connected 3-manifold containing a solid torus S such that there is a homeomorphism h of S3-Int(T) onto M-Int(S). It is known [I ] that M is S' if T is unknotted or if T has the type of a trefoil knot. In [5] it is asserted that M is SI in every case. The proof given in [51 shows only that h must take some longitude of T onto a longitude of S-a fact that is not sufficient to prove this assertion. The purpose of this paper is to provide a proof that M is S3 in the case that T has the type of a torus knot. We first give a canonical description of the torus knot of type (p, q). We consider 5' as the one point compactification of E3 which we represent in cylindrical coordinates (r, 0, z). Let R be the torus R= { (r, 0, z): (r-2)2+z2= 1}. Let p and q be relatively prime positive integers. We assume, for convenience, that p>q; since it is known that the torus knots of types (p, q) and (q, p) are equivalent. For 1 <k?p let X=h(3, 2k2r/p, 0) and yk= (1, 2kir/p, 0). Let ak be an arc on R from xk to yk+4 that lies in the positive z half space and which increases monotonically with 0. (It is to be understood that the subscripts are to be reduced modulo p.) Let bk be the arc on R from Xk+q to yk+q that lies in the negative z half space and the plane O=2(k+q)7r/p. We denote the simple closed curve UL, (akUbk) by Kp,. Figure 1 shows K6,3.