Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T06:53:48.705Z Has data issue: false hasContentIssue false
  • English
  • Français

On Prime Immersions of S1 into R2

Published online by Cambridge University Press:  20 November 2018

John R. Martin
John R. Martin*
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A C1-mapping ƒ from the oriented circle S1 into the oriented plane R2 such that f f’ (t) ≠ 0 for all t is called a regular immersion. We call a point p in Im f a double point if f-1(p) is a two element set with the corresponding tangent vectors being linearly independent. A regular immersion which is one-to-one except at a finite number of points whose images are double points is called a normal immersion. The work of Whitney [7], Titus [3] and Verhey [6] shows that the normal immersions form a dense open subset in the space of regular immersions with the usual C1-topology, and can be characterized up to diffeomorphic equivalence by a combinatorial invariant called the intersection sequence.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 589 - 593
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Martin, J. R., Determining knot types from diagrams of knots, Pacific J. Math. 51 (1974), 241–249.Google Scholar
2. Marx, M. L., A combinatorial invariant that characterizes normal immersions of S1 into R2, Duke Math. J. 41 (1974), 145–149.Google Scholar
3. Titus, C. J., A theory of normal curves and some applications, Pacific J. Math. 10 (1960), 1083–1096.Google Scholar
4. Treybig, L. B., A characterization of the double point structure of the projection of a polygonal knot in regular position, Trans. Amer. Math. Soc. 130 (1968), 223–247.Google Scholar
5. Treybig, L. B., Prime mappings, Trans. Amer. Math. Soc. 130 (1968), 248–253.Google Scholar
6. Verhey, R. F., Diffeomorphic invariants of immersed circles, Trans. Amer. Math. Soc. 163 (1972), 47–63.Google Scholar
7. Whitney, Hassler, On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284.Google Scholar