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

 

The number of knot group representations
is not a Vassiliev invariant


Author: Michael Eisermann
Journal: Proc. Amer. Math. Soc. 128 (2000), 1555-1561
MSC (1991): Primary 57M25
Published electronically: October 5, 1999
MathSciNet review: 1657727
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Abstract: For a finite group $G$ and a knot $K$ in the $3$-sphere, let $F_G(K)$ be the number of representations of the knot group into $G$. In answer to a question of D.Altschuler we show that $F_G$ is either constant or not of finite type. Moreover, $F_G$ is constant if and only if $G$ is nilpotent. We prove the following, more general boundedness theorem: If a knot invariant $F$ is bounded by some function of the braid index, the genus, or the unknotting number, then $F$ is either constant or not of finite type.


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Additional Information

Michael Eisermann
Affiliation: Mathematisches Institut der Universität Bonn, Beringstr.1, 53115 Bonn, Germany
Email: eiserm@math.uni-bonn.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05287-9
PII: S 0002-9939(99)05287-9
Received by editor(s): July 9, 1998
Published electronically: October 5, 1999
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society