Markov classes in certain finite quotients of Artin’s braid group. (English) Zbl 0621.20025

Authors’ summary: This paper studies three finite quotients of the sequence of braid groups \(\{B_ n\); \(n=1,2,...\}\). Each has the property that Markov classes in \(B_{\infty}=\amalg B_ n\) pass to well-defined equivalence classes in the quotient. We are able to solve the Markov problem in two of the quotients, obtaining canonical representatives for Markov classes and giving a procedure for reducing an arbitrary representative to the canonical one. The results are interpreted geometrically, and related to link invariants of the associated links and the value of the Jones polynomial [V. Jones, Bull. Am. Math. Soc., New. Ser. 12, 103-111 (1985; Zbl 0564.57006)] on the corresponding classes.
Reviewer: S.C.Althoen


20F36 Braid groups; Artin groups
20F65 Geometric group theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)


Zbl 0564.57006
Full Text: DOI


[1] Baumslag, G., Automorphism groups of residually finite groups, J. London Math. Soc., 28, 117-118 (1961) · Zbl 0124.26003
[2] D. Bennequin,Entrelacements et equations de Pfaff, Thesis de Doctorat D’Etat, Univ. Paris VII, 1982.
[3] J. Birman,Braids, Links and Mapping class groups, Annals of Math. Studies #82, Princeton Univ. Press, 1973.
[4] V. F. R. Jones,Braid groups, Hecke algebras and type II_1factors, preprint.
[5] Jones, V. F. R., A polynomial invariant for knots via von Neumann algebras, Bull. Am. Math. Soc., 12, 103-111 (1985) · Zbl 0564.57006
[6] W. B. R. Lickorish and K. Millett,Some evaluations of link polynomials, preprint. · Zbl 0607.57003
[7] Morton, H., Threading knot diagrams, Math. Proc. Camb. Phil. Soc., 99, 247-260 (1986) · Zbl 0595.57007
[8] H. Murakami,A recursive calculation of the Arf invariant, preprint. · Zbl 0605.57003
[9] B. Wajnryb,Presentation of the symplectic group over a field of 3 elements, preprint. · Zbl 0533.57002
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