The faithfulness question for the Burau representation

Moody, John

doi:10.1090/S0002-9939-1993-1158006-X

The faithfulness question for the Burau representation
HTML articles powered by AMS MathViewer

by John Moody

Proc. Amer. Math. Soc. 119 (1993), 671-679

DOI: https://doi.org/10.1090/S0002-9939-1993-1158006-X

PDF | Request permission

Abstract:

We consider the Burau and Gassner representations of the braid groups ${B_n}$. A sufficient condition for faithfulness, involving just a pair of arcs, is shown to be necessary as well for all but at most two values of $n$. In the Burau case, this implies nonfaithfulness for $n \geqslant 10$.

References

J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, DOI 10.1090/S0002-9947-1928-1501429-1
Wilhelm Magnus, Collected papers, Springer-Verlag, New York, 1984. Edited and with introductory material by Gilbert Baumslag and Bruce Chandler. MR 725504
D. D. Long, A note on the normal subgroups of mapping class groups, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 79–87. MR 809501, DOI 10.1017/S0305004100063957

Über die freié Äquivalenz geschlossener Zöpfe

1

Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281

A lemma on systems of knotted curves

9

D. D. Long, On the linear representation of braid groups, Trans. Amer. Math. Soc. 311 (1989), no. 2, 535–560. MR 943606, DOI 10.1090/S0002-9947-1989-0943606-3
Toshitake Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 339–363. MR 975088, DOI 10.1090/conm/078/975088
John Atwell Moody, The Burau representation of the braid group $B_n$ is unfaithful for large $n$, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 379–384. MR 1098347, DOI 10.1090/S0273-0979-1991-16080-5
Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
John McCarthy, A “Tits-alternative” for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), no. 2, 583–612. MR 800253, DOI 10.1090/S0002-9947-1985-0800253-8