Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 946435
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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] Gérald Bourgeois and David 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-B 286 (1978), no. 18, A771–A773 (French, with English summary). MR 497662
  • [2] Ralph H. Fox, A remarkable simple closed curve, Ann. of Math. (2) 50 (1949), 264–265. MR 0030745
  • [3] John Milnor, Lectures on the ℎ-cobordism theorem, Notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J., 1965. MR 0190942
  • [4] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. MR 0515288
  • [5] David Spring, Proper homotopy and immersion theory, Topology 11 (1972), 295–305. MR 0295378

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M25, 57M99

Retrieve articles in all journals with MSC: 57M25, 57M99

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society