Given \(\textbf{n}=(n_{1},\ldots ,n_{r})\in \mathbb {N}^r\) , let \(\Gamma _{\textbf{n}}\) be a group presentable as \(\begin{aligned} \left\langle \gamma _{1},\ldots ,\gamma _{r}\,|\,\gamma _{1}^{n_{1}}=\gamma _{2}^{n_{2}}=\cdots =\gamma _{r}^{n_{r}}\right\rangle . \end{aligned}\) If \(\gcd (n_i,n_j)=1\) for all \(i\not =j\) , we say \(\Gamma _{\textbf{n}}\) is a generalized torus knot group and otherwise say it is a generalized torus link group. This definition includes torus knot and link groups ( \(r=2\) ), that is, fundamental groups of the complement of a torus knot or link in \(S^{3}\) . Let G be a connected complex reductive affine algebraic group. We show that the G-character varieties of generalized torus knot groups are path-connected. We then count the number of irreducible components of the \({{\,\mathrm{\textrm{SL}}\,}}(2,\mathbb {C})\) -character varieties of \(\Gamma _\textbf{n}\) when \(n_i\) is odd for all i.
