No Access

A GATHERING PROCESS IN ARTIN BRAID GROUPS

Omsk Branch of Institute of Mathematics, (Siberian Branch of Russian Academy of Science), Pevtsova st. 13, Omsk, 644099, Russia

Search for more papers by this author

and

ILYA V. KAZACHKOV

Omsk Branch of Institute of Mathematics, (Siberian Branch of Russian Academy of Science), Pevtsova st. 13, Omsk, 644099, Russia

Search for more papers by this author

https://doi.org/10.1142/S0218196706003190Cited by:0 (Source: Crossref)

Abstract

In this paper we construct a gathering process by the means of which we obtain new normal forms in braid groups. The new normal forms generalize Artin–Markoff normal forms and possess an extremely natural geometric description. In the two last sections of the paper we discuss the implementation of the introduced gathering process and the questions that arose in our work. This discussion leads us to some interesting observations, in particular, we offer a method of generating a random braid.

Keywords:

AMSC: 20F36

References

E. Artin, Ann. Math. 48(1), 101 (1947), DOI: 10.2307/1969218. Crossref, Web of Science, Google Scholar
A. V. Borovik, A. G. Myasinkov and V. N. Remeslennikov, Int. J. Algebra Comput. 13(6), 705 (2003), DOI: 10.1142/S0218196703001596. Link, Web of Science, Google Scholar
N. Bourbaki , Groupes et Algèbres de Lie , 2nd edn. ( Hermann , 1968 ) . Google Scholar
J. Debois and S. Nechaev, J. Statist. Phys. 88, 2767 (1997). Google Scholar
P. Dehornoy , Braids and Self Distributivity , Progress in Mathematics 192 ( Birkhauser , 2000 ) . Crossref, Google Scholar
P. Dehornoy, Cryptography on Braids, Contemp. Math. to appear . Google Scholar
D. B. A. Epstein et al. , Word Processing in Groups ( Jones and Bartlett , Boston, London , 1992 ) . Crossref, Google Scholar
I. Kapovichet al., J. Algebra 264(2), 665 (2003), DOI: 10.1016/S0021-8693(03)00167-4. Crossref, Web of Science, Google Scholar
G. S. Makanin, Proc. Russ. Acad. Sci. 182(3), (1968). Google Scholar
A. A. Markoff, Tr. Mat. Inst. Steklova 16, 53 S (1945). Google Scholar
A. Vershik, S. Nechaev and R. Bikbov, Commun. Math. Phys. 212, 469 (2000), DOI: 10.1007/s002200000221. Crossref, Web of Science, Google Scholar
V. V. Vershinin, Adv. Math. 54(2), 3 (1999). Google Scholar