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

Dedicated to Peter E. Newstead, on his $80^{\text{th}}$ birthday. 19 pages, 3 figures