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Cacti, braids and complex polynomials. (English) Zbl 0976.57004
Summary: The study of the topological classification of complex polynomials began in the 19th century by J. Lüroth [Math. Ann. 3, 181-184 (1871; JFM 03.0192.02)], A. Clebsch [ibid. 6, 216-230 (1873; JFM 05.0285.03)] and A. Hurwitz [ibid. 39, 1-61 (1891; JFM 23.0429.01)]. In the works of S. Zdravkovska and A. G. Khovanskij [Usp. Mat. Nauk 25, No. 4(154), 179-180 (1970; Zbl 0201.25801); J. Knot Theory Ramifications 5, No. 1, 55-75 (1996; Zbl 0865.57001)] the problem is reduced to a purely combinatorial one, namely the study of a certain action of the braid groups on a class of tree-like figures that we, following I. P. Goulden and D. M. Jackson [Eur. J. Comb. 13, No. 5, 357-365 (1992; Zbl 0804.05023)], call “cacti”. Using explicit computation of the braid group orbits, enumerative results of Goulden and Jackson [loc. cit.], and also establishing some combinatorial invariants of the action, we provide the topological classification of polynomials of degree up to 9 (previous results were known up to degree 6).

57M12 Low-dimensional topology of special (e.g., branched) coverings
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F36 Braid groups; Artin groups
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