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Polynomial invariants of pseudo-Anosov maps. (English) Zbl 1268.57002

In [Topology 34, No. 1, 109–140 (1995; Zbl 0837.57010)], M. Bestvina and M. Handel gave an algorithmic proof of Thurston’s classification theorem for mapping classes (see e.g., the volume [Travaux de Thurston sur les surfaces. Seminaire Orsay. Astérisque, No. 66–67. Paris: Société Mathématique de France. 284 p. (1979; Zbl 0406.00016)]. If \([F]\) is a pseudo-Anosov map acting on an orientable surface \(S\), their algorithm allows to construct a graph \(G\) (homotopic to \(S\) when \(S\) is punctured), a suitable map \(f: G \to G\) (called train track map) and the associated transition matrix \(T\) (whose Perron-Frobenius eigenvalue is the dilatation of \([F]\); see [F. R. Gantmacher, The theory of matrices, Vol. 2. Providence, RI: AMS Chelsea Publishing. (1959; Zbl 0927.15002)]).
The dilatation \(\lambda(F)\) is an invariant of the conjugacy class \([F]\) in the modular group of \(S\), studied by C. T. McMullen [Ann. Sci. Éc. Norm. Supér. (4) 33, No. 4, 519–560 (2000; Zbl 1013.57010)] and in several subsequent papers.
The present paper introduces a new approach to the study of invariants of \([F]\), when \([F]\) is pseudo-Anosov: starting from the Bestvina-Handel algorithm, the authors investigate the structure of the characteristic polynomial of the transition matrix \(T\) and obtain two new integer polynomials (both containing \(\lambda(F)\) as their largest real root), which turn out to be invariants of the given pseudo-Anosov mapping class.
The degrees of these new polynomials, as well as of their product, are invariants of [F], too; simple formulas are given for computing them by a counting argument from an invariant train track.
The paper gives also examples of genus 2 pseudo-Anosov maps having the same dilatation, which are distinguished by the new invariants.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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References:

[1] DOI: 10.1016/0040-9383(94)E0009-9 · Zbl 0837.57010
[2] DOI: 10.1080/10586458.2000.10504648 · Zbl 0982.57005
[3] DOI: 10.1515/9781400839049
[4] Fathi A., Travaux de Thurston sur les Surfaces 66 (1979)
[5] Gantmacher F. R., The Theory of Matrices (1959) · Zbl 0085.01001
[6] McMullen C., Ann. Sci. École Norm. Sup. 33 pp 519– · Zbl 1013.57010
[7] Penner R. C., Combinatorics of Train Tracks (1992) · Zbl 0765.57001
[8] Rolfsen D., Knots and Links (1990) · Zbl 0854.57002
[9] Rykken E., Michigan Math. J. 46 pp 281–
[10] DOI: 10.1090/S0273-0979-1988-15685-6 · Zbl 0674.57008
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