Commentarii Mathematici Helvetici

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Volume 74, Issue 4, 1999, pp. 530–547
DOI: 10.1007/s000140050104

Published online: 1999-12-31

Boundary slopes of knots

M. Culler[1] and P. B. Shalen[2]

(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