Myers, Robert2025-01-152025-01-151982-09Myers, R. (1982). Simple knots in compact, orientable 3-manifolds. Transactions of the American Mathematical Society, 273(1), pp. 75-91. https://doi.org/10.1090/S0002-9947-1982-0664030-00002-9947https://hdl.handle.net/20.500.14446/345770A 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 ∂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 K(M) of knot groups, i.e, the set of groups π₁, (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 S³ if and only if every simple closed curve in M shrinks to a point inside a connected sum of graph manifolds and 3-cells. © 1982 American Mathematical Society.application/pdfThis material has been previously published. In the Oklahoma State University Library's institutional repository this version is made available through the open access principles and the terms of agreement/consent between the author(s) and the publisher. The permission policy on the use, reproduction or distribution of the material falls under fair use for educational, scholarship, and research purposes. Contact Digital Resources and Discovery Services at lib-dls@okstate.edu or 405-744-9161 for further information.10.1090/S0002-9947-1982-0664030-0Articleapplied mathematicsnumerical and computational mathematicspure mathematicsmathematical sciences3-manifoldknotsimple knotsimple 3-manifoldsemisimple 3-manifoldhyperbolic 3-manifoldknot groupPoincaré ConjectureORCID: 0009-0008-5271-4140 (Myers, R)ScopusID: 7403700679 (Myers, R)1088-6850