Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-23T20:52:50.136Z Has data issue: false hasContentIssue false

Infinitely Periodic Knots

Published online by Cambridge University Press:  20 November 2018

Erica Flapan*
Affiliation:
Rice University, Houston, Texas
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.

One aspect of the study of 3-manifolds is to determine what finite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action? In the case of periodic transformations of homology spheres, Smith [18] classified the types of fixed point sets which could occur. For homology 3-spheres the fixed point set will be ∅, S0, S1, or S2. Fox [4] looked at periodic transformations of the three sphere which leave a knot invariant and, using Smith's classification of fixed point sets, determined that there were eight types of transformations according to how the fixed point set met the knot. For convenience we shall say a knot is (a, b)-periodic if there is a periodic transformation of S3 leaving the knot invariant with fixed point set homeomorphic to a and with the fixed point set meeting the knot in a set homeomorphic to b.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Burde, G. and Zieschang, H., Eine Kennzeichnung der Torus knoten, Math. Ann. 167 (1966), 169176.Google Scholar
2. Conner, P. E., Transformation groups on a K(π, 1) II, Michigan Math. J. 6 (1959), 413417.Google Scholar
3. Fox, R. H., On the imbedding of polyhedra in 3-space, Ann. of Math. 49 (1948), 462470.Google Scholar
4. Fox, R. H., Knots and periodic transformations, Proc. The Univ. of Georgia Inst. (Prentice-Hall, Englewood Cliffs, N.J., 1961), 120167.Google Scholar
5. Freedman, M., Haas, J. and Scott, P., Lease area incompressible surfaces in 3-manifolds, to appear in Inventiones Mathematicae.Google Scholar
6. Giffen, C. H., On transformations of the 2-sphere fixing in a knot, Bull. Amer. Math. Soc. 73 (1967), 913914.Google Scholar
7. Hartley, R. I., Knots and involutions, Math. Z. 171, (1980), 175185.Google Scholar
8. Hartley, R. I., Knots with free period, Can. J. Math. 33 (1981), 91102.Google Scholar
9. Jaco, W., Lectures on three-manifold topology, Memoirs AMS 43 (1980).CrossRefGoogle Scholar
10. Jaco, W. and Shalen, P., Seifert fibered spaces in 3-manifolds, Memoirs AMS (1979).CrossRefGoogle Scholar
11. Johannson, K., Homotopy equivalences of 3-man if olds with boundaries, Lecture Notes in Mathematics 761 (1979).CrossRefGoogle Scholar
12. Meeks, W., A survey of the geometric results in the classical theory of minimal surfaces, Bol. Soc. Bras. Mat. 12 (1981), 2986.Google Scholar
13. Meeks, W. and Scott, P., Finite group actions on 3-manifolds, preprint.Google Scholar
14. Murasugi, K., On periodic knots, Comment. Math. Helv. 46 (1971), 162174.Google Scholar
15. Myers, R., Companionship of knots and the Smith conjecture, Trans. Amer. Math. Soc. 259 (1980), 132.Google Scholar
16. Seifert, H., Topologie dreidimensionalen gefaserter Raume, Acta Math. 60 (1933), 147238.Google Scholar
17. Simon, J., An algebraic classification of knots in S3 , Ann. of Math. 97 (1973), 113.Google Scholar
18. Smith, P. A., Transformations of finite period II, Ann. of Math. 40 (1939), 690711.Google Scholar
19. Swarup, G. A., P. A. Smith conjecture for cable knots, Quart. J. Math. Oxford 31 (1980), 105108.Google Scholar
20. Waldhausen, F., Gruppen mit Zentrum und dreidimensionale Mannigfaltigkeiten, Topology 6 (1967), 505517.Google Scholar
21.Proceedings of the 1979 Conference on the Smith Conjecture at Columbia University, to appear.Google Scholar