Stratification of 𝑆𝑈(𝑟)-character varieties of twisted Hopf links

Abstract

We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with n n twists, namely Γ n = ⟨ x , y | [ x n , y ] = 1 ⟩ {\Gamma }_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle into the group S U ( r ) SU(r) . For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case r = 2 r = 2 , we provide a complete geometric description of the character variety, proving that this S U ( 2 ) SU(2) -character variety is a deformation retract of the larger S L ( 2 , C ) SL(2,\mathbb {C}) -character variety, as conjectured by Florentino and Lawton. In the case r = 3 r = 3 , we also describe different strata of the S U ( 3 ) SU(3) -character variety according to the semi-simple type of the representation.