|
Organizers |
Symplectic and unitary quotients of Burau representation, and 3-move and t_3, bar t_4-move conjectures
by
Jozef H. Przytycki
George Washington University
We present general ideas which can lead to a solution of Montesinos-Nakanishi conjecture and its generalizations. In particular we speculate about the geometrical meaning of the finite quotients of the braid group described by J.Assion (Symplectic and Unitary cases). Coxeter showed that Cn = Bn/(\sigmai)3 is finite iff n≤ 5. Assion found two basic cases in which Cn/(Ideal) is finite: ``Symplectic and Unitary" cases. We noticed with B.Westbury (motivated by Q.Chen) simple presentations for Assion ideals (compare also Murasugi). Let \Delta5 = (\sigma1\sigma2\sigma3\sigma4)5 be a generator of the center of the braid group, B5. (1) ``Symplectic case". Cn/(\Delta10) is a finite group. (2) ``Unitary case". Cn/(\Delta15) is a finite group. We should remark that \Delta30=1 in C5, and C5/\Delta5 is a simple group PSp(4, 3) (projective symplectic group).
Our approach to "moves" conjectures is via the following very concrete problem: (i) The 5-tangle yielded by the 5-braid \Delta10 is 3-equivalent to the trivial 5-braid tangle. (ii)] The 5-tangle yielded by the 5-braid \Delta15 is t3, t̅4 equivalent to the trivial 5-braid tangle.
Date received: January 23, 2000
Copyright © 2000 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caea-17.