A volume form on the SU(2)-representation space of knot groups (Q5907043)

Let \(M\setminus V(K)\) be the exterior of a knot \(K\) in \(S^3\) and \(G_K=\pi_1 M\) its group. An irreducible representation \(\rho:G_K\rightarrow \text{SU}(2)\) is called regular when \(H_\rho^1(M)\cong \mathbb{R}\) in which \(H_\rho^1(M)= H^1(M, \operatorname{Ad}\rho)\) is the twisted cohomology group of \(M\) with coefficients in the representation \( \operatorname {Ad} \rho:G_K\rightarrow \Aut(\text{su}(2))\), Ad being the adjoint representation. Using a decomposition of \(M\) similar to the Heegaard splitting used in the construction of the Casson invariant, the author proves that \(\operatorname{Reg}(K)\) carries a well-defined \(1\)-volume form. In fact, this volume form can be interpreted as a Reidemeister torsion, and an explicit computation of this form is given for torus knots of type \((2,q)\).