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On the injectivity of the Vassiliev homomorphism of singular Artin monoids

Published online by Cambridge University Press:  17 April 2009

Noelle Antony
Noelle Antony
Affiliation:
School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia.
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I prove general combinatorial properties which apply to singular Artin monoids and examine their relationship with the Vassiliev homomorphism ≠. I show that ≠ preserves the Intermediate Property, discovered by Corran, which holds in positive singular Artin monoids of finite type. From this it follows that ≠ is injective for a class of monoids which include singular Artin monoids of type I2(p), generalising a result of East.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 3 , December 2004 , pp. 401 - 422
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Artin, E., ‘Theorie der Zöpfe’, Hamburg Abh. 4 (1925), 47–72.CrossRefGoogle Scholar
[2]Baez, J., ‘Link invariants of finite type and perturbation theory’, Lett. Math. Phys. 26 (1992), 43–51.CrossRefGoogle Scholar
[3]Birman, J.S., ‘New points of view in knot theory’, Bull. Amer. Math. Soc. (N.S.) 28 (1993), 253–286.CrossRefGoogle Scholar
[4]Brieskorn, E. and Saito, K., ‘Artin-Gruppen und Coxeter-Gruppen’, Invent. Math. 17 (1972), 245–271.CrossRefGoogle Scholar
[5]Charney, R., ‘Artin groups of finite type are biautomatic’, Math. Ann. 292 (1992), 671–683.CrossRefGoogle Scholar
[6]Corran, R., ‘A normal form for a class of monoids including the singular braid monoid’, J. Algebra 223 (2000), 256–282.CrossRefGoogle Scholar
[7]Corran, R., On monoids related to braid groups, Ph.D. Thesis (School of Mathematics and Statistics, University of Sydney, 2000).Google Scholar
[8]Dasbach, O.T. and Gemein, B., ‘A faithful representation of the singular braid monoid on three strands’, Ser. Knots Everything 24 (2000), 48–58.Google Scholar
[9]East, J., ‘Birman's conjecture is true for I 2(p)’, (preprint) http://www.maths.usyd.edu.au:8000/u/pubs/publist/preprints/2002/east-8.html.Google Scholar
[10]Fenn, R., Rolfsen, D. and Zhu, J., ‘Centralizers in the braid group and the singular braid monoid’, Enseign. Math. 42 (1996), 75–96.Google Scholar
[11]Garside, F.A., ‘On the braid group and other groups’, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235–254.CrossRefGoogle Scholar
[12]Godelle, E. and Paris, L., ‘On singular Artin monoids’, http://math.u-bourgogne.fr/topolog/IMB2-publication.html année 2003, n.356.Google Scholar
[13]González-Meneses, E., ‘Presentations for the monoids of singular braids on closed surfaces’, Comm. Algebra 30 (2002), 2829–2836.CrossRefGoogle Scholar
[14]Humphreys, J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29 (Cambridge Univ. Press. Cambridge, U.K., 1990).CrossRefGoogle Scholar
[15]Járai, A. Jr, ‘On the monoid of singular braids’, Topology Appl. 96 (1999), 109–119.CrossRefGoogle Scholar
[16]Paris, L., ‘Artin monoids inject on their groups’, Comment. Math. Helv. 77 (2002), 609–637.CrossRefGoogle Scholar
[17]Paris, L., ‘The proof of Birman's conjecture on singular braid monoids’, (preprint) http://math.u.-bourgogne.fr/topolog/IMB2-publication.html année 2003, n.338 ArXiv:math.GR/0306422 v1.Google Scholar
[18]Vassiliev, V., Cohomology of knot spaces, (Arnold, V.I., Editor), Theory of Singularities and its Applications 1 (Amer. Math. Soc., Profidence, R.I., 1990).Google Scholar
[19]Zhu, J., ‘On singular braids’, J. Knot Theory Ramifications 6 (1997), 427–440.CrossRefGoogle Scholar