Montesinos Amilibia, José María and González Acuña, Francisco Javier (1980) Nonamphicheiral codimension 2 knots. Canadian Journal of Mathematics-Journal Canadien de Mathématiques, 32 (1). pp. 185-194. ISSN 0008-414X
Restricted to Repository staff only until 31 December 2020.
Official URL: http://cms.math.ca/10.4153/CJM-1980-014-x
An n-knot (Sn+2,Sn) is said to be amphicheiral if there is an orientation-reversing autohomeomorphism of Sn+2 which leaves Sn invariant as a set. An n-knot is said to be invertible if there is an orientation-preserving autohomeomorphism of Sn+2 whose restriction to Sn is an orientation-reversing autohomeomorphism of Sn. The authors prove that for any integer n there are smooth n-knots which are neither amphicheiral nor invertible. Actually, they prove it for n≥2, referring to the paper of H. F. Trotter [Topology 2 (1963), 275–280; errata, MR 30, p. 1205] for the case n=1. The methods employed are mainly algebraic, involving for example the duality pairings of R. C. Blanchfield and J. Levine, and in most cases the work of previous authors is used to guarantee the existence of knots with the desired algebraic properties.
|Uncontrolled Keywords:||knots of codimension which are neither amphicheiral nor invertible; non- symmetric Alexander polynomial; knot cobordism group; fibred knot|
|Subjects:||Sciences > Mathematics > Topology|
S. Akbulut and R. Kirby, An exotic involution of S 4 (preprint).
E. Becerra, Simple homotopy equivalent knot complements, Ph.D. Thesis (1976), Instituto de Matemâticas de la U.N.A.M., Mexico.
S. S. Cappell, Superspinning and knot complement, in Topology of manifolds (Markham Publishing Co., 1970), 358-383.
S. S. Cappell and J. L. Shaneson, Topological knots and knot cobordism, Topology 12 (1973), 33-40.
M. M. Cohen, A course in simple homotopy theory, Springer (1970).
R. H. Crowell, The Group G'/G" of a knot group, Duke Math. J. 30 (1963), 349-354.
M . S. Farber, Linking coefficients and two dimensional knots, Soviet Math. Dokl. 16 (1975), 647-650.
R. H. Fox, Some problems in knot theory, in Topology of o-manifolds and related topics (Prentice Hall, N.J., 1962), 168-176.
L. Fuchs, Infinite abelian groups (Academic Press 86,1970).
P. J. Hilton and S. Wylie, Homology theory (Cambridge University Press, 1960).
C. Kearton, Noninvertible knots of codimension 2, Proc. Am. Math. Soc. 40 (1973), 274-276.
M. Kervaire, Les noeuds de dimensions supérieures, Bull. Soc. Math. France 03 (1965), 225-271.
On higher dimensional knots, Differential and Combinatorial Topology, A symposium in honor of Marston Morse, Princeton Univ. Press (1965), 105-119.
R. C. Kirby and L. C. Siebenmann, Codimension two locally flat imbeddings, Notices Amer.Math. Soc. 18 (1971), 983.
E. Landau, Vorlesungen uber Zahlentheorie, Verlag bon S. Hirzel Leipzig (1974).
J. Levine, Knot cobordism groups in codimension two, Comm. Math. Helv. 44 (1969), 229-244.
Knot modules, Annals of Math Studies 84 and Trans. Amer. Math. Soc. 229 (1977), 1-50.
D. W. Sumners, Homotopy torsion in codimension two knots, Proc. Amer. Math. Soc. 24(1970), 229-240.
Polynomial invariants and the integral homology of coverings of knots and links, Inventiones Math. 15 (1972), 78-90.
H. F. Trotter Non-invertible knots exist, Topology 2 (1963), 275-280.
B. L. Van der Waerden, Moderne algebra (Springer, 1950).
C. T. C. Wall, Classification problems in differential topology—VI, Topology 6 (1967), 273-296.
T. Yanagawa, On ribbon 2-knots, Osaka j . Math. 6 (1969), 447-464.
E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495.
|Deposited On:||29 Nov 2012 10:43|
|Last Modified:||29 Nov 2012 10:43|