Abstract

This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, $P_{n}(n > 3)$. Long and Paton proved that if a Burau matrix $M$ has ones on the diagonal and zeros below the diagonal then $M$ 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 $P_{n}$ under the reduced Gassner representation is $n-1$, 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.

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