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HlyU upregulates expression of the haemolysin, HlyA, of Vibrio cholerae. DNA sequence analysis indicates that HlyU is an 11.9 kDa protein containing a putative helix-turn-helix motif and belonging to a family of small regulatory proteins, including NoIR (Rhizobium meliloti), SmtB (Synechococcus PCC 7942) and ArsR (plasmids R773, Escherichia coli; pI258, Staphylococcus aureus; and pSX267, Staphylococcus xylosus). An hlyU mutant was constructed by insertional inactivation, and found to be deficient in the production of both the haemolysin and a 28 kDa secreted protein. The mutant was assessed for virulence in the infant mouse cholera model, revealing a 100-fold increase in the LD50. This suggests that HlyU promotes expression of virulence determinant(s) in vivo.

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.

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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].