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Transactions of the American Mathematical Society

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

 
 

 

Proper knot theory in open $ 3$-manifolds


Authors: Peter Churchyard and David Spring
Journal: Trans. Amer. Math. Soc. 308 (1988), 133-142
MSC: Primary 57M25; Secondary 57M99
DOI: https://doi.org/10.1090/S0002-9947-1988-0946435-9
MathSciNet review: 946435
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Abstract: This paper introduces a theory of proper knots, i.e., smooth proper embeddings of $ {{\mathbf{R}}^1}$ into open $ 3$-manifolds. Proper knot theory is distinguished by the fact that proper isotopies of knots are not ambient in general. A uniqueness theorem for proper knots is proved in case the target manifold is the interior of a one-dimensional handlebody.


References [Enhancements On Off] (What's this?)

  • [1] G. Bourgeois and D. Spring, Groupes d'homotopie de l'espace des plongements propres d'une $ n$-variété ouverte dans une $ m$-variété ouverte, C. R. Acad. Sci. Paris Sér. A 286 (1978), 771-773. MR 497662 (80b:58021)
  • [2] R. H. Fox, A remarkable simple closed curve, Ann. of Math. (2) 50 (1949), 264-265. MR 0030745 (11:45b)
  • [3] J. Milnor, Lectures on the $ h$-cobordism theorem, Princeton Univ. Press, 1965. MR 0190942 (32:8352)
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  • [5] D. Spring, Proper homotopy and immersion theory, Topology 11 (1971), 295-305. MR 0295378 (45:4444)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0946435-9
Article copyright: © Copyright 1988 American Mathematical Society

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