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.
Cite this article
M. Culler, P. B. Shalen, Boundary slopes of knots. Comment. Math. Helv. 74 (1999), no. 4, pp. 530–547DOI 10.1007/S000140050104