The number of knot group representations is not a Vassiliev invariant

Abstract

For a finite group G G and a knot K K in the 3 3 -sphere, let F G ( K ) F_G(K) be the number of representations of the knot group into G G . In answer to a question of D. Altschuler we show that F G F_G is either constant or not of finite type. Moreover, F G F_G is constant if and only if G G is nilpotent. We prove the following, more general boundedness theorem: If a knot invariant F F is bounded by some function of the braid index, the genus, or the unknotting number, then F F is either constant or not of finite type.