Periodic Orbits of a Dynamical System Related to a Knot

Following [SW2] we consider a knot group G, its commutator subgroup K = [G,G], a finite group Σ and the space Hom(K,Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σx of Hom(K,Σ) onto itself: σxρ(a) = ρ(xax) ∀a ∈ K, ρ ∈ Hom(K,Σ). As proven in [SW1], the dynamical system (Hom(K,Σ), σx) is a shift of finite type. In the case when Σ is abelian, Hom(K,Σ) is finite. In this paper we calculate the periods of orbits of (Hom(K,Z/p), σx) where p is prime in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.