The article under review studies character varieties of torus knots. Given a knot \(K_{m,n}\subset S^3\), the fundamental group of the complement \(S^3-K_{m,n}\) has the finite presentation \(\Gamma_{m,n}= \langle x, y|x^n = y^m\rangle\). The space of representations of \(\Gamma_{m,n}\) in a group \(G\), modulo conjugation by \(G\), defines a GIT quotient \(X(\Gamma_{m,n},G)\) which can be identified with the variety of characters of the representations, or \(G\)-character variety of \(\Gamma_{m,n}\).
This paper addresses G-character varieties for \(G = \mathrm{SL}(r,\mathbb{C})\), \(\mathrm{GL}(r,\mathbb{C})\) and \(\mathrm{PGL}(r,\mathbb{C})\). When \(G\) is a matrix group, the representations of \(\Gamma_{m,n}\) are given by pairs of matrices \(A\), \(B\), such that \(A^n = B^m\). In the cases of this article, the study of the character variety relies on the study of the simultaneous conjugacy classes for \(A\) and \(B\), which allows to stratify the character variety (c.f. section 5) accordingly with the representations being reducible or not. Corollary 5.5 shows that these character varieties are connected and, in section 6, it is calculated the number of maximal dimensional irreducible components.
Section 7 gives a full description of the three character varieties for r = 2, recovering previous results but using this geometric method of stratifications. These descriptions are used in section 8 to prove Theorem 1.1 describing the \(\mathrm{SL}(3,\mathbb{C})\)-character variety of \(\Gamma_{m,n}\) and calculate its K-theory and the Euler characteristic. In section 9 it is described the closure of each strata in the description. In section 10 the analogous results are proved for \(\mathrm{GL}(3,\mathbb{C})\) and \(\mathrm{PGL}(3,\mathbb{C})\).