Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Abstract:
This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, . Long and Paton proved that if a Burau matrix has ones on the diagonal and zeros below the diagonal then is the identity matrix. In this paper, a generalization of Long and Paton's result will be proved. Our main theorem is that if the trace of the image of an element of under the reduced Gassner representation is , then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.