Character Varieties of Generalized Torus Knot Groups

Abstract

Given $\mathbf{n}=(n_1,\dots ,n_r)\in {\mathbb{N}}^r$, let $\hat{I}{“}_{\mathbf{n}}$ be a group presentable as $$\left\langle γ_{1},\ldots,γ_{r}\:|\:γ_{1}^{n_{1}}=γ_{2}^{n_{2}}=\cdots=γ_{r}^{n_{r}}\right\rangle. $$ If $\gcd (n_i,n_j)=1$ for all $i\ne j$, we say $\hat{I}{“}_{\mathbf{n}}$ is a {\it generalized torus knot group} and otherwise say it is a {\it 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 $\mathrm{SL}(2,\mathbb{C})$-character varieties of $\hat{I}{“}_{\mathbf{n}}$ when $n_i$ is odd for all $i$.

21 pages, 2 figures, accepted for publication in the Mediterranean Journal of Mathematics