Non-Amphicheiral Codimension 2 Knots

González-Acuña, F. ; Montesinos, José M.

Canadian journal of mathematics, Tome 32 (1980) no. 1, pp. 185-194

Résumé

An n-knot (Sn+2, Sn) is amphicheiral if there is an orientation reversing autohomeomorphism of Sn+2 leaving Sn invariant as a set. It is invertible if there is an orientation preserving autohomeomorphism of Sn+2 whose restriction to Sn is an orientation reversing autohomeomorphism of Sn onto itself. In 1961 Fox [8, Problem 35] asked if there exist non-amphicheiral locally flat 2-knots. We will prove the following THEOREM 1. For any integer n there are smooth n-knots which are neither amphicheiral nor invertible.A knot (Sn+2, Sn) is + amphicheiral (resp. —amphicheiral) if there is an orientation reversing autohomeomorphism f of Sn+2 leaving Sn invariant such that f| Snpreserves (resp. reverses) orientation.

Détail

DOI : 10.4153/CJM-1980-014-x

Bibliographie

[1] 1. Akbulut, S. and Kirby, R., An exotic involution of S4 (preprint). Google Scholar

[2] 2. Becerra, E., Simple homotopy equivalent knot complements, Ph.D. Thesis (1976), Instituto de Matemâticas de la U.N.A.M., Mexico. Google Scholar

[3] 3. Cappell, S. S., Superspinning and knot complement, in Topology of manifolds (Markham Publishing Co., 1970), 358–383. Google Scholar

[4] 4. Cappell, S. S. and Shaneson, J. L., Topological knots and knot cobordism, Topolog. 12 (1973), 33–40. Google Scholar

[5] 5. Cohen, M. M., A course in simple homotopy theory, Springer (1970). Google Scholar

[6] 6. Crowell, R. H., The Group G'/G” of a knot group, Duke Math. J. 30 (1963), 349–354. Google Scholar

[7] 7. Farber, M. S., Linking coefficients and two dimensional knots, Soviet Math. Dokl. 16 (1975), 647–650. Google Scholar

[8] 8. Fox, R. H., Some problems in knot theory, in Topology of o-manifolds and related topics (Prentice Hall, N.J., 1962), 168–176. Google Scholar

[9] 9. Fuchs, L., Infinite abelian groups (Academic Press 36, 1970). Google Scholar

[10] 10. Hilton, P. J. and Wylie, S., Homology theory (Cambridge University Press, 1960). Google Scholar

[11] 11. Kearton, C., Noninvertible knots of codimension 2, Proc. Am. Math. Soc. 40 (1973), 274–276. Google Scholar

[12] 12. Kervaire, M., Les noeuds de dimensions supérieures, Bull. Soc. Math. Franc. 03 (1965), 225–271. Google Scholar

[13] 13. Kervaire, M., On higher dimensional knots, Differential and Combinatorial Topology, A symposium in honor of Marston Morse, Princeton Univ. Press (1965), 105–119. Google Scholar

[14] 14. Kirby, R. C. and Siebenmann, L. C., Codimension two locally flat imbeddings, Notices Amer. Math. Soc. 18 (1971), 983. Google Scholar

[15] 15. Landau, E., Vorlesungen uber Zahlentheorie, Verlag bon S. Hirzel Leipzig (1974). Google Scholar

[16] 16. Levine, J., Knot cobordism groups in codimension two, Comm. Math. Helv. 44 (1969), 229–244. Google Scholar

[17] 17. Levine, J., Knot modules, Annals of Math Studies 84 and Trans. Amer. Math. Soc. 229 (1977), 1–50. Google Scholar

[18] 18. Sumners, D. W., Homotopy torsion in codimension two knots, Proc. Amer. Math. Soc. 24 (1970), 229–240. Google Scholar

[19] 19. Sumners, D. W., Polynomial invariants and the integral homology of coverings of knots and links, Inventiones Math. 15 (1972), 78–90. Google Scholar

[20] 20. Trotter, H. F. Non-invertible knots exist, Topolog. 2 (1963), 275–280. Google Scholar

[21] 21. Van der Waerden, B. L., Moderne algebra (Springer, 1950). Google Scholar

[22] 22. Wall, C. T. C., Classification problems in differential topology—VI, Topolog. 6 (1967), 273–296. Google Scholar

[23] 23. Yanagawa, T., On ribbon 2-knots, Osaka j . Math. 6 (1969), 447–464. Google Scholar

[24] 24. Zeeman, E. C., Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495. Google Scholar