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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Simple knots in compact, orientable $ 3$-manifolds

Author: Robert Myers
Journal: Trans. Amer. Math. Soc. 273 (1982), 75-91
MSC: Primary 57N10; Secondary 57M25, 57M40
MathSciNet review: 664030
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Abstract: A simple closed curve $ J$ in the interior of a compact, orientable $ 3$-manifold $ M$ is called a simple knot if the closure of the complement of a regular neighborhood of $ J$ in $ M$ is irreducible and boundary-irreducible and contains no non-boundary-parallel, properly embedded, incompressible annuli or tori. In this paper it is shown that every compact, orientable $ 3$-manifold $ M$ such that $ \partial M$ contains no $ 2$-spheres contains a simple knot (and thus, from work of Thurston, a knot whose complement is hyperbolic). This result is used to prove that such a $ 3$-manifold is completely determined by its set $ \mathcal{K}(M)$ of knot groups, i.e, the set of groups $ {\pi _1}(M - J)$ as $ J$ ranges over all the simple closed curves in $ M$. In addition, it is proven that a closed $ 3$-manifold $ M$ is homeomorphic to $ {S^3}$ if and only if every simple closed curve in $ M$ shrinks to a point inside a connected sum of graph manifolds and $ 3$-cells.

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Keywords: $ 3$-manifold, knot, simple knot, simple $ 3$-manifold, semisimple $ 3$-manifold, hyperbolic $ 3$-manifold, knot group, Poincaré Conjecture
Article copyright: © Copyright 1982 American Mathematical Society