Irreducible representations of the braid group B3B_3 in dimension 6

We use $q$-Pascal's triangle to define a family of representations of dimension 6 of the braid group $B_3$ on three strings. Then we give a necessary and sufficient condition for these representations to be irreducible

B3B_3 block representations of dimension 6 and braid reversions

We construct a family of six dimensional block representations of the braid group $B_3$ on three strings. We show that these representations can be used to distinguish braids of knots with 9 and 10 crossings from their reversed braids

The Interplay between Linear Representations of the Braid Group

We consider Wada's representation as a twisted version of the standard action of the braid group, Bn, on the free group with n generators. Constructing a free group, Gnm, of rank nm, we compose Cohen's map Bn→Bnm and the embedding Bnm→Aut(Gnm) via Wada's map. We prove that the composition factors of the obtained representation are one copy of Burau representation and m−1 copies of the standard representation after changing the parameter t to tk in the definitions of the Burau and standard representations. This is a generalization of our previous result concerning the standard Artin representation of the braid group

Generalizations of the standard Artin representation are unitary

We consider the Magnus representation of the image of the braid group under the generalizations of the standard Artin representation discovered by M. Wada. We show that the images of the generators of the braid group under the Magnus representation are unitary relative to a Hermitian matrix. As a special case, we get that the Burau representation is unitary, which was known and proved by C. C. Squier

On the considerations adopted by Breidze and Traczyk towards the faithfulness of Burau representation for n=4n=4

This work discusses the open problem of the faithfulness of the reduced Burau representation for $n=4$. Birman showed that in order to prove this representation is faithful, it is sufficient to find two matrices $A$ and $B$ that generate a free group of rank 2. Breidze and Traczyk proved that $A^{3}$ and $B^{3}$ generate the free group of rank 2. In our work, we show that $A^{2}$ and $B^{2}$ generate the free group of rank 2

Krammer's Representation of the Pure Braid Group,

We consider Krammer's representation of the pure braid group on three strings: 3→(3,[±1,±1]), where and are indeterminates. As it was done in the case of the braid group, 3, we specialize the indeterminates and to nonzero complex numbers. Then we present our main theorem that gives us a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of Krammer's representation of the pure braid group, 3