JournalscmhVol. 74 , No. 4DOI 10.1007/s000140050104

Boundary slopes of knots

  • M. Culler

    University of Illinois at Chicago, USA
  • P. B. Shalen

    University of Illinois at Chicago, USA
Boundary slopes of knots cover

Abstract

The results in this paper show that simple connectivity of a 3-manifold is reflected in the behavior of essential surfaces in exteriors of knots in the manifold. A corollary of the main theorem is that any non-trivial knot, with irreducible complement, in a homotopy 3-sphere must have two boundary slopes that differ by at least 2. This statement is false for knots in a homology 3-sphere. The main theorem itself applies more generally to knots in closed orientable 3-manifolds with cyclic fundamental group.