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Non-Amphicheiral Codimension 2 Knots

Published online by Cambridge University Press:  20 November 2018

F. González-Acuña  and
José M. Montesinos
F. González-Acuña
Affiliation:
Instituto de Matemáticas de la U.N.A.M., Mexico
José M. Montesinos
Affiliation:
The Institute for Advanced Study, Princeton, New Jersey
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 185 - 194
Copyright
Copyright © Canadian Mathematical Society 1980

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