Character Varieties of Generalized Torus Knot Groups

Abstract

Given n=(n1,…,nr)∈Nr, let Γn be a group presentable as (Formula presented.) If gcd(ni,nj)=1 for all i≠j, we say Γn is a generalized torus knot group and otherwise say it is a 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 S3. 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 SL(2,C)-character varieties of Γn when ni is odd for all i.