Flawedness Conjecture for Generalized Torus Link Groups

Determine whether every generalized torus link group Γ_n = ⟨γ_1, …, γ_r | γ_1^{n_1} = γ_2^{n_2} = ··· = γ_r^{n_r}⟩ is flawed; that is, establish that for every connected complex reductive affine algebraic group G with maximal compact subgroup K, the G-character variety X_{Γ_n}(G) admits a strong deformation retraction onto X_{Γ_n}(K).

Background

Generalized torus link groups are defined by presentations of the form Γ_n = ⟨γ_1, …, γ_r | γ_1^{n_1} = γ_2^{n_2} = ··* = γ_r^{n_r}⟩, encompassing torus knot groups (when gcd(n_i, n_j) = 1 for all i ≠ j) and torus link groups otherwise.

A finitely generated group Γ is called G-flawed if the G-character variety X_Γ(G) strongly deformation retracts onto X_Γ(K), where G is a connected complex reductive affine algebraic group and K ≤ G is a maximal compact subgroup. Γ is flawed if it is G-flawed for all such G. Prior work shows this property for abelian and free groups, and for certain torus knot cases (e.g., SU(2)/SL(2, C) and some SL(3, C) instances).

This paper develops structural and topological results for character varieties of generalized torus link groups (e.g., path-connectedness, description via exact sequences involving free products of finite cyclic groups), which provide evidence toward the conjecture but do not resolve it in full generality.