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ON CYCLES AND COVERINGS ASSOCIATED TO A KNOT

LILYA LYUBICH

Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794-3660, USA

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and

MIKHAIL LYUBICH

Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794-3660, USA

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https://doi.org/10.1142/S0218216513500740Cited by:0 (Source: Crossref)

Abstract

Let 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

is finite, i.e. it consists of several cycles. In [Periodic orbits of a dynamical system related to a knot, J. Knot Theory Ramifications20(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

Keywords:

AMSC: 57M25, 57M10, 15A24

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