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On the character variety of periodic knots and links

Published online by Cambridge University Press:  17 January 2001

HUGH M. HILDEN ,
MARÍA TERESA LOZANO  and
JOSÉ MARÍA MONTESINOS-AMILIBIA
HUGH M. HILDEN
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822, U.S.A.
MARÍA TERESA LOZANO
Affiliation:
Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
JOSÉ MARÍA MONTESINOS-AMILIBIA
Affiliation:
Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
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Abstract

For a hyperbolic knot, the excellent component of the character curve is the one containing the complete hyperbolic structure on the complement of the knot. In this paper we explain a method to compute the excellent component of the character variety of periodic knots. We apply the method to those knots obtained as the preimage of one component of a 2-bridge link by a cyclic covering of S3 branched on the other component. We call these knots periodic knots with rational quotient. Among this class of knots are the ‘Turk's head knots’. Finally we give some invariants deduced from the excellent component of the character curve, such as the h-polynomial and the limit of hyperbolicity for all the periodic knots with rational quotient, up to 10 crossings, which are not 2-bridge or toroidal.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 3 , November 2000 , pp. 477 - 490
Copyright
© 2000 Cambridge Philosophical Society

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