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Transactions of the American Mathematical Society

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

 
 

 

On knots with nontrivial interpolating manifolds


Author: Jonathan Simon
Journal: Trans. Amer. Math. Soc. 160 (1971), 467-473
MSC: Primary 55.20
DOI: https://doi.org/10.1090/S0002-9947-1971-0288753-3
MathSciNet review: 0288753
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Abstract: If a polygonal knot $ K$ in the $ 3$-sphere $ {S^3}$ does not separate an interpolating manifold $ S$ for $ K$, then $ S - K$ does not carry the first homology of either closed component of $ {S^3} - S$. It follows that most knots $ K$ with nontrivial interpolating manifolds have the property that a simply connected manifold cannot be obtained by removing a regular neighborhood of $ K$ from $ {S^3}$ and sewing it back differently.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0288753-3
Keywords: Knot, $ 2$-manifold, $ 3$-manifold, interpolating manifold, Property P$ $, homotopy $ 3$-sphere, cube-with-holes, cube-with-handles, Poincaré conjecture
Article copyright: © Copyright 1971 American Mathematical Society

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