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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Deformations of dihedral representations


Authors: Michael Heusener and Eric Klassen
Journal: Proc. Amer. Math. Soc. 125 (1997), 3039-3047
MSC (1991): Primary 57M25, 57M05
MathSciNet review: 1443155
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Abstract: G. Burde proved (1990) that the $\mathrm {SU}_2 (\Bbb {C})$ representation space of two-bridge knot groups is one-dimensional. The same holds for all torus knot groups. The aim of this note is to prove the following:
Given a knot $\k \subset S^3$ we denote by $\Hat C_2$ its twofold branched covering space. Assume that there is a prime number $p$ such that $H_1(\Hat C_2,\Bbb {Z}_p)\cong \ZZ_p$. Then there exist representations of the knot group onto the binary dihedral group $D_p \subset \mathrm {SU}_2 (\Bbb {C})$ and these representations are smooth points on a one-dimensional curve of representations into $\mathrm {SU}_2 (\Bbb {C})$.


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Additional Information

Michael Heusener
Affiliation: Uni–GH–Siegen\ Fachbereich Mathematik\ Hölderlinstraße 3\ 57068 Siegen\ Germany
Email: heusener@hrz.uni-siegen.d400.de

Eric Klassen
Affiliation: Department of Mathematics\ Florida State University\ Tallahassee\ Florida 32306
Email: klassen@math.fsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04195-6
PII: S 0002-9939(97)04195-6
Keywords: Knot groups, group representations, $\SU$
Received by editor(s): September 7, 1993
Additional Notes: The second author was supported in part by a National Science Foundation Postdoctoral Research Fellowship.
Communicated by: Ronald Stern
Article copyright: © Copyright 1997 American Mathematical Society