Mahler measure, links and homology growth

Abstract

Let l be an oriented link of d components with nonzero Alexander polynomial ∆(u 1 , . . . , u d ). Let Λ be a finite-index subgroup of H 1 (S 3 −l) ∼ = Z d , and let M Λ be the corresponding abelian cover of S 3 branched along l. The growth rate of the order of the torsion subgroup of H 1 (M Λ ), as a suitable measure of Λ approaches infinity, is equal to the Mahler measure of ∆.1. Introduction. Associated to any knot k ⊂ S 3 is a sequence of Alexander polynomials ∆ i , i ≥ 1, in a single variable such that ∆ i+1 divides ∆ i . Likewise, for any oriented link of d components there is a sequence of Alexander polynomials in d variables. Following the usual custom, we refer to the first Alexander polynomial of a knot or a link as the Alexander polynomial, and we denote it simply by ∆.In [Go] C. McA. Gordon examined the homology groups of r-fold cyclic covers M r of S 3 branched over a knot k. He proved that when each zero of the Alexander polynomial ∆ of k has modulus one (and hence is a root of unity), the finite values of |H 1 (M r )| are periodic in r. Gordon conjectured that when some zero of ∆ is not a root of unity, the finite values of |H 1 (M r )| grow exponentially. More than fifteen years later two independent proofs of Gordon's conjecture, one by R. Riley [Ri] and another by F. Gonzaléz-Acuña and H. Short [GoSh], appeared. Both employed the Gel'fond-Baker theory of linear forms in the logarithms of algebraic integers [Ba], [Ge].We extend the above results for knots, replacing the term "finite values of |H 1 (M r )|" with "order of the torsion subgroup of H 1 (M r )," while at the same time proving a general result for links in S 3 . Our proof, which is motivated by [SiWi2], identifies the torsion subgroup of the homology of a finite abelian branched cover with the connected components of periodic points in an associated algebraic dynamical system. Theorem 21.1 of [Sc], an enhanced version of a theorem of D. Lind, K. Schmidt and T. Ward [LiScWa], then completes our argument.Recognizing that relatively few topologists are familiar with algebraic dynamical systems, we have endeavored to make this paper self-contained. The reader who desires to know more about such dynamical systems is encouraged to consult the extraordinary monograph [Sc].

Metabelian SL(n, ℂ) representations of knot groups

“…This is a consequence of Theorems 3.10 and 3.12, which give more precise information on conjugacy classes of irreducible metabelian representations for knots with nontrivial Alexander polynomial. Our proofs make use of the extensive work on homology groups of n-fold branched covers of knots that began with [Gordon 1972] and was continued in [Riley 1990;González-Acuña and Short 1991;Silver and Williams 2002], and our existence results of representations are strengthenings of previous ones (compare [Klassen 1991, Corollary 11]). …”

Three Dimensions of Knot Coloring

“…Why not extend our palette of colors to the entire circle? This is the main idea of [14], [15]. The set of Fox T-colorings of a knot turns out to be a compact abelian group, and conjugation in the knot group by a meridian induces a homeomorphism.…”

Exponential Torsion Growth for Random 3-Manifolds

“…First results on the relation of this growth rate to the (logarithmic) Mahler measure of the Alexander polynomial of the knot or link can be found in [Ri90] and [GS91]. Equality of this growth rate and the Mahler measure of the Alexander polynomial are due to Silver and Williams [SW02a], and extensions of these results and an interpretation in the context of ℓ 2 -invariants can be found in [SW02b] as well as in the more recent papers [BV13,Ra12,Le14].…”