The extended Alexander group of an oriented virtual link l of d components is defined. From its abelianization a sequence of polynomial invariants Δi (u1,…,ud, v), i=0, 1,…, is obtained. When l is a classical link, Δi reduces to the well-known ith Alexander polynomial of the link in the d variables u1v,…,udv; in particular, Δ0 vanishes.
Let G be a finitely generated group, and let λ ∈ G. If there exists a knot k such that πk = π 1 (S 3 \ k) can be mapped onto G sending the longitude to λ, then there exists infinitely many distinct prime knots with the property. Consequently, if πk is the group of any knot (possibly composite), then there exists an infinite number of prime knots k 1 , k 2 , · · · and epimorphisms · · · → πk 2 → πk 1 → πk each perserving peripheral structures. Properties of a related partial order on knots are discussed.
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].
Properties of polynomial invariants ∆ i for oriented virtual links are established. The effects of taking mirror images and reversing orientation of the link diagram are described.The relationship between ∆ 0 (u, v) and an invariant of F. Jaeger, L. Kauffman, H. Saleur and J. Sawollek is discussed.
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