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High-dimensional knots with $ \pi\sb 1\cong{\bf Z}$ are determined by their complements in one more dimension than Farber's range


Author: William Richter
Journal: Proc. Amer. Math. Soc. 120 (1994), 285-294
MSC: Primary 57Q45
MathSciNet review: 1195730
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Abstract: The surgery theory of Browder, Lashof, and Shaneson reduces the study of high-dimensional smooth knots $ {\Sigma ^n} \to {S^{n + 2}}$ with $ {\pi _1} \cong \mathbb{Z}$ to homotopy theory. We apply Williams's Poincaré embedding theorem to a highly connected Seifert surface. Then such knots are determined by their complements if the $ \mathbb{Z}$-cover of the complement is $ [(n + 2)/3]$-connected; we improve Farber's work by one dimension.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1195730-8
Keywords: Knots with $ {\pi _1} \cong \mathbb{Z}$, Poincaré embeddings, unstable normal invariant
Article copyright: © Copyright 1994 American Mathematical Society