The Number of Knot Group Representations Is Not a Vassiliev Invariant

For a finite group G and a knot K in the 3-sphere, let FG(K) be the number of representations of the knot group into G. In answer to a question of D.Altschuler we show that FG is either constant or not of finite type. Moreover, FG 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.