Zeros of the Alexander polynomial of knot.(English)Zbl 1133.57009

For a knot $$K$$ in the $$3$$-sphere, let $$X_r(K)$$ denote the $$r$$-fold cyclic branched covering of the $$3$$-sphere over $$K$$. In the study of the periodicity of the order of $$H_1(X_r(K))$$, C. McA. Gordon [Trans. Am. Math. Soc. 168, 357–370 (1972; Zbl 0238.55001)] showed that if the Alexander polynomial of $$K$$ has a root which is not a root of unity then the finite values of $$| H_1(X_r(K))|$$ are unbounded. R. Riley [Bull. Lond. Math. Soc. 22, No. 3, 287–297 (1990; Zbl 0727.57002)] and F. González Acuña [Rev. Mat. Univ. Complutense Madr. 4, No.1, 97–120 (1991; Zbl 0756.57001)] independently improved this by showing that the finite values of $$| H_1(X_r(K))|$$ grow exponentially in $$r$$. Riley used $$p$$-adic analysis, and González Acuña made use of the fact that the growth is equal to the Mahler measure of the Alexander polynomial. In particular, the latter suggested the possibility of interpreting the growth as the entropy of a dynamical system. The paper under review first remarks that the dual group of $$H_1(X_\infty(K))$$, where $$X_\infty(K)$$ is the infinite cyclic cover of the knot complement, is a solenoid whose dimension is equal to the degree of the Alexander polynomial. A solenoid is a compact connected finite-dimensional abelian group. By the computation of the entropy of an automorphism of a solenoid due to D.A. Lind and T. B. Ward [Ergodic Theory Dyn. Syst. 8, No. 3, 411–419 (1988; Zbl 0634.22005)], the main theorem claims that the entropy of the meridian action $$t_p$$ on the $$p$$-adic coefficient Alexander module is the sum of $$\log| \alpha_i| _p$$ over all zeros of the Alexander polynomial satisfying $$| \alpha_i| _p>1$$, where $$| \cdot| _p$$ denotes the $$p$$-adic norm, and that the entropy of dual action of meridian on the dual of $$H_1(X_\infty(K))$$ is the sum of the entropies of $$t_p$$ over $$p\leq \infty$$. As a corollary, the growth can be expressed as the sum of the $$p$$-adic entropies. This recovers the above result by Riley and González Acuña. Moreover, it is shown that if the (usual) Alexander module is finitely generated then the entropies of $$t_p$$ are zero for all finite primes $$p$$. Finally, the leading coefficient of the Alexander polynomial is expressed as the sum of the entropies of $$t_p$$ for finite primes $$p$$.

MSC:

 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 11S05 Polynomials 37B40 Topological entropy
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References:

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