Abstract
<p align="left">The faithfulness of the Burau representation of the 4-strand braid group, $B_4$, remains an open question.<br />In this work, there are two main results. First, we specialize the indeterminate $t$ to a complex number on the unit circle, and we find a necessary condition for a word of $B_4$ to belong to the kernel of the representation. Second, by using a simple algorithm,<br />we will be able to exclude a family of words in the generators from belonging to the kernel of the reduced Burau representation.</p>
Highlights
Magnus and Peluso (1969) showed that the Burau representation is faithful for n ≤ 3. Moody (1991) showed that it is not faithful for n ≥ 9; this result was improved to n ≥ 6 by Long and Paton (1992)
We specialize the indeterminate t to a complex number on the unit circle, and we find a necessary condition for a word of B4 to belong to the kernel of the representation
By using a simple algorithm, we will be able to exclude a family of words in the generators from belonging to the kernel of the reduced Burau representation
Summary
Magnus and Peluso (1969) showed that the Burau representation is faithful for n ≤ 3. Moody (1991) showed that it is not faithful for n ≥ 9; this result was improved to n ≥ 6 by Long and Paton (1992). Magnus and Peluso (1969) showed that the Burau representation is faithful for n ≤ 3. Moody (1991) showed that it is not faithful for n ≥ 9; this result was improved to n ≥ 6 by Long and Paton (1992). The non-faithfulness for n = 5 was shown by Bigelow (1999). The question of whether or not the Burau representation for n = 4 is faithful is still open. I=1 we find the general form of the words an and bn and we prove that they are not in the kernel of the representation for any non-zero natural number n. We conclude that there is no word of such forms in the kernel of the representation
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