Irreducible representations of knot groups into \(\mathrm{SL}(n,\mathbf{C})\) (Q2405868)

The title of this paper characterizes the main result. The authors prove that under certain conditions involving an \(\alpha \in \mathbb {C}^*\) the group \(\Gamma\) of a knot \(K\) in a homology sphere admits a representation \(\varrho _\lambda\) (where \(\lambda ^n = \alpha \in \mathbb {C}^*\)) of \(\Gamma\) in \(\mathrm{SL}(n,\mathbb C)\) having the following properties: \quad 1) \(\varrho _\lambda\) is reducible and metabelian \quad 2) \(\varrho _\lambda\) is a smooth point in the variety \(R_n (\Gamma)\) of representations \quad 3) \(\varrho _\lambda\) is contained in a unique \(n^2 + n - 2\) dimensional component \(R_{\varrho _\lambda}\) of \(R^n (\Gamma)\) \quad 4) \(R_{\varrho _\lambda}\) contains irreducible non-abelian representations which deform \(\varrho _\lambda\). The ``certain conditions'' cited above are that the \((t - \alpha)\) torsion of the Alexander module of \(K\) is cyclic and of the form \(\mathbb {C}[t^{\pm1}]/(t - \alpha)^{n-1}, n \geq 2\). \quad The authors also establish a condition for such a representation to be ``upgradable'' to a representation in \(\mathrm{SL}(n + 1, \mathbb C)\). \quad As heavy use is made of cohomological considerations in the proofs, cohomological properties of \(\Gamma\) are discussed at some length. \quad Several examples are included.