No Access

ON THE CONJUGACY PROBLEM OF POSITIVE BRAIDS

E. A. ELRIFAI

Department of Mathematics, Faculty of Sciences, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

Search for more papers by this author

and

M. BENKHALIFA

Department of Mathematics, Faculty of Sciences, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

Search for more papers by this author

https://doi.org/10.1142/S0218216504003160Cited by:2 (Source: Crossref)

Abstract

In the symmetric group S_n, we introduced the notion of crossing and linking numbers to each permutation. Then a unique factorization of a permutation is given due to its crossing number of its factors and how the factors are linked. Consequently we introduced a matrix associated to each permutation, which we used it as a tool to prove that positive braids with different matrices are not conjugate braids. Up to n≤5 it is proved that two positive permutation braids are conjugate if and only if they have the same matrix, and a complete calculation of these matrices with the associated link type is given.

Keywords:

AMSC: Primary: 20F36, Secondary: 20E45, Secondary: 57M27

References

J. S. Birman, Ann. Math. Stud. 82, (1974). Google Scholar
A. Kawauchi , A Survey of Knot Theory ( Birkhäuser Verlag , 1996 ) . Google Scholar
E. A. Elrifai and H. R. Morton, Quart. J. Math. Oxford 45(2), 479 (1994). Crossref, Web of Science, Google Scholar
J. Stallings, Proc. Symp. Pure Maths 32, 55 (1978). Web of Science, Google Scholar
J. S. Birman and R. F. Williams, Topology 22, 47 (1983). Crossref, Web of Science, Google Scholar
E. A. Elrifai, Positive braids and Lorenz links, Ph.D. thesis, Liverpool University (1988) . Google Scholar
C. Mc A. Gordon et al. , Knots in Hellas "98" , Proc. Int. Conf. on Knot Theory and Its Ramification , Series on Knots and Everythings 24 ( World Scientific , 2000 ) . Google Scholar
H. R. Morton, low-dimensional topology, London Math. Soc., Lecture Note Ser. 95 (Cambridge Univ. Press, 1985) pp. 86–105. Crossref, Google Scholar
D. Rolfsen , Knots and Links ( Publish or Perish Press , Berkeley, CA , 1976 ) . Google Scholar