Character Varieties of Generalized Torus Knot Groups

Abstract

Given $\mathbf{n}=(n_1,\dots ,n_r)\in {\mathbb{N}}^r$, let ${\Gamma }_{\mathbf{n}}$ be a group presentable as $$⟨{\gamma }_1,\dots ,{\gamma }_r|{\gamma }_1^{n_1}={\gamma }_2^{n_2}=\cdots ={\gamma }_r^{n_r}⟩.$$ If $\gcd (n_i,n_j)=1$ for all $i\ne j$, we say ${\Gamma }_{\mathbf{n}}$ 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 ${\Gamma }_{\mathbf{n}}$ when $n_i$ is odd for all $i$.

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

References in corpus (6)

• The topology of moduli spaces of free group representations
• Topology of character varieties of Abelian groups
• Singularities of free group character varieties
• Geometry of the SL(3,C)-character variety of torus knots
• Flawed groups and the topology of character varieties
• Geometry of $\mathrm{SU}(3)$-character varieties of torus knots