DOI: 10.12732/ijam.v38i3.7
A NOTE ON THE FAITHFULNESS OF
THE EVALUATED GASSNER REPRESENTATION
OF THE PURE BRAID GROUP
Mohamad N. Nasser
Department of Mathematics and Computer Science
Beirut Arab University
P.O. Box 11-5020, Beirut, LEBANON
Abstract. Burau representation of the braid group, Bn, has been proved to be faithful for \leq 3 and unfaithful for n \geq 5; whereas the case n = 4 remains open. On the other hand, the question of faithfulness of Gassner representation of the pure braid group, Pn, is still open for n \geq 4. For any n \geq 4, T. Chuna specified a family of new non-trivial elements in the kernel of the evaluated Burau representation at m^{th} root of unity, where $m\in \mathbb{N}^*-\{3\}$. Along the same lines as Chuna’s work, we find, for n \geq 4, a family of new non-trivial elements in the kernel of the evaluated Gassner representation at m^{th} root of unity, where $m\in \mathbb{N}^*-\{3\}$. This result may possibly be a road toward answering the open question of the faithfulness of Gassner representation.
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DOI: 10.12732/ijam.v38i3.7
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2025
Volume: 38
Issue: 3
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