On cycles and coverings associated to a knot

Let Κ be a knot, G be the knot group, K be its commutator subgroup, and x be a distinguished meridian. Let Σ be a finite abelian group. The dynamical system introduced by Silver and Williams in [Augmented group systems and n-knots, Math. Ann. 296 (1993) 585-593; Augmented group systems and shifts of finite type, Israel J. Math. 95 (1996) 231-251] consisting of the set Hom(K, Σ) of all representations ρ: K → Σ endowed with the weak topology, together with the homeomorphism σ_x: Hom} (K, Σ) → Hom(K,Σ), σ_xρ(a) = ρ (xax^-1) ∀ a ∈ K, ρ ∈ Hom (K, Σ), is finite, i.e. it consists of several cycles. In [Periodic orbits of a dynamical system related to a knot, J. Knot Theory Ramifications 20(3) (2011) 411-426] we found the lengths of these cycles for Σ = ℤ/p,p is prime, in terms of the roots of the Alexander polynomial of the knot, mod p. In this paper we generalize this result to a general abelian group Σ. This gives a complete classification of depth 2 solvable coverings over S³\Κ.