Based on a joint work with Yoshikazu Yamaguchi (Tokyo Tech). In this talk, we will mainly focus on the trace-free characters of SL(2,C)-representations of a knot group. This gives an algebraic subset of the character variety of the knot group with a symmetry in terms of the metabelian characters. This symmetry is useful to understand a relationship to the character variety of 2-fold branched cover of 3-sphere over the knot.