Symmetry of Reidemeister torsion on -representation spaces of knots

We study two sorts of actions on the space of conjugacy classes of irreducible SU 2 -representations of a knot group. One of them is an involution which comes from the algebraic structure of SU 2 and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on a 1-dimensional smooth part of the space, which is canonically oriented and metrized via a Reidemeister torsion volume form. As an application we show that the Reidemeister torsion function on the 1-dimensional subspace has symmetry about the metrization.