Commentarii Mathematici Helvetici
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Published online: 1999-12-31
Boundary slopes of knotsM. Culler and P. B. Shalen (1) University of Illinois at Chicago, USA
(2) University of Illinois at Chicago, USA
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.
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Culler M., Shalen P.: Boundary slopes of knots. Comment. Math. Helv. 74 (1999), 530-547. doi: 10.1007/s000140050104