Tangent cones and local geometry of the representation and character varieties of knot groups (Q969648)

Let \(\Gamma\) be a classical knot group and \(R(\Gamma)= \text{Hom}(\Gamma,\text{SL}_2(\mathbb{C}))\) a representation variety. The paper contains an explicit calculation of the tangent cone to \(R(\Gamma)\) at an Abelian representation \(\rho_\lambda\) corresponding to a root \(\alpha= \lambda^2\) of the Alexander polynomial of the knot where the \((t-\alpha)\)-torsion of the Alexander module is cyclic of order \(r= 2\); in case of \(r\geq 3\) an approximation to the tangent cone by a descending sequence of cones included in the Zariski tangent space is described. Simultaneously the local structure of \(R(\Gamma)\) near the reducible non-Abelian representation \(\phi_\lambda\) with the same character as \(\rho_\lambda\) is studied. Explicit results are presented for \(\dim_{\mathbb{C}}Z^1(\Gamma,{\mathfrak s}{\mathfrak l}_{\phi_\lambda})\).