A new zeta function, natural for links. (English) Zbl 0851.58037

Williams, R. F.

Hirsch, M. W. (ed.) et al., From topology to computation: Proceedings of the Smalefest. Papers presented at the conference “From topology to computation: Unity and diversity in the mathematical sciences” held at the University of California at Berkeley, USA, August 5-9, 1990 in honor of Stephen Smale’s 60th birthday. New York: Springer-Verlag. 270-278 (1993).

Let \(A\) be an \(n \times n\) matrix with coefficients in a (not necessarily abelian) group \(F\). The author introduces two related determinant-like functions, link-det \((I-A)\) and cycle-det\((I-A)\). He first defines a free-knot symbol to be an equivalence class under cyclic permutations of a nonvanishing product \(A(i_1, i_2), A(i_2, i_3) \cdots A(i_k, i_1)\), for a sequence of nodes \(i_1, i_2, \cdots, i_k\), with \(i_r \neq i_s\) when \(r \neq s\). The first function is link-det\((I-A) = \sum^* (-1)^I x_1 x_2 \cdots x_I\), where the sum \(\sum^*\) is over free-link symbols, i.e., products of free-knot symbols, no two of which have a node in common. The determinant-like functions have nice properties, suitable to applications to the case where \(A\) is a Markov matrix arising from a dynamical system. For example, the characteristic polynomial corresponding to cycle-det \((I-A)\) satisfies the Cayley-Hamilton equation [S. Kennedy, M. Stafford and the author, “A new Cayley-Hamilton theorem”, to appear]. As a corollary, one obtains the identity, essential in symbolic dynamics, \(\text{exp}(\sum_{i=0}^\infty - \text{tr} (A^i)/i) = \text{ link-det }(I-A)\). Link-det \((I - PAP^{-1}) = \text{ link-det }(I-A)\) whenever \(P\) is a permutation matrix (the link-determinant is not necessarily invariant under the general linear group). Finally, link-det \((I-A)\) becomes \(\text{det }(I - A)\) under abelianization. Some proofs in this article are only valid for matrices \(A\) obtained from a geometrical Lorenz attractor with a double saddle connection [see, e.g., the author, Publ. Math., Inst. Hautes Etudes Sci. 50, 73-99 (1979; Zbl 0484.58021)]; the proofs in the general case follow from results in the above reference to Kennedy, Stafford and the author. Results on the links in these geometrical Lorenz attractors are obtained.
For the entire collection see [Zbl 0779.00016].

Reviewer: V.Baladi (MR 94k:58116)

MSC:

37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
57M25	Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

zeta function; Markov matrix; links; Lorenz attractors

Citations:

Zbl 0484.58021