The Picard group, the figure-eight knot group and J groups
by
Hiroki Sato
Department of Mathematics, Faculty of Science, Shizuoka University

In this talk we will state that the Picard group G_P is a two-generator group and a Jørgensen group. Furthermore we will draw a fundamental polyhedoron for the Picard group G_P and describe a complete set of relations for G_P as a two-generator group.

Here we consider two-generator groups G_{ik, \sigma} = <A, B_{ik, \sigma} > generated by

A = æ ç è 1 1 0 1 ö ÷ ø and Bik, \sigma = æ ç è ik\sigma -k2\sigma- 1/\sigma \sigma ik\sigma ö ÷ ø ,

where k in R and \sigma in C\{0}.

THEOREM (i) The Picard group G_P is conjugate to G_{1/2, \pi/2}, that is, G_P = RG_{1/2, \pi/2}R^-1, where

R = æ ç è 1 i/2 0 1 ö ÷ ø

(ii) The following relations form a complete set of relations for G_P:

(B-1ABA2BAB-1A2B-1ABA2BAB-1AB)2 = 1

(B-1ABA2BAB-1A2B-1ABA)2 = 1

(AB-1ABA2BAB-1A2B-1ABA2BAB-1AB)2 = 1

(AB-1ABA2BAB-1A2B-1ABA)2 = 1

(B-1ABA)3 = 1

(AB-1ABA)2 = 1

(AB-1ABA2B-1ABA2BAB-1A2B-1ABA2BAB-1AB)2 = 1

(AB-1ABA2B-1ABA2BAB-1A2B-1ABA)3 = 1,

where B = RB_{1/2, \pi/2}R^-1.

COROLLARY. The Picard group is a two-generator group and a Jørgensen group.

Date received: June 11, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-03.