Two Dimensional Representations of the Braid Group B3 and its Burau Representation Squared

We construct a family of six dimensional block representations of the braid group $B_3$ on three strings. We show that some of these representations can be used to detect braid vertibility of some known knots and others of 9 and 10 crossings.

We give a complete classification of simple representations of the braid group B_3 with dimension $\leq 5$ over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation $\rho: B_3 \to GL(V)$ is determined up to isomorphism by the eigenvalues $\lambda_1, \lambda_2, ..., \lambda_d$ of the image of the generators for d=2,3 and a choice of a $\delta=\sqrt{\det \rho(\sigma_1)}$ for d=4 or a choice of $\delta=\sqrt[5]{\det \rho(\sigma_1)}$...

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.

We show that the Lawrence--Krammer representation can be obtained as the quantization of the symmetric square of the Burau representation. This connection allows us to construct new representations of braid groups

In this paper the author finds explicitly all finite-dimensional irreducible representations of a series of finite permutation groups that are homomorphic images of Artin braid group.

In this paper we study irreducible matrix representations of the welded braid group $WB_n$, also known as the group of conjugating automorphisms of a free group $F_n.$ We prove that $WB_n$ has no irreducible representations of dimension $r,$ where $2\leqslant r\leqslant n-2$ for $n\geqslant 5.$ We give complete classification of all extensions of irreducible representations of the braid group $B_n$ to the welded braid group $WB_n$ of dimensions $n-1$ (for $n\geqslant 7$) and ...

We give a method to produce representations of the braid group $B_n$ of $n-1$ generators ($n\leq \infty$). Moreover, we give sufficient conditions over a non unitary representation for being of this type. This method produces examples of irreducible representations of finite and infinite dimension.

We consider subgroups of the braid groups which are generated by $k$-th powers of the standard generators and prove that any infinite intersection (with even $k$) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau's representations of the braid groups at roots of unity. Furthermore, we show that the image of the braid group on 3 strands by these representations is either a finite group, for a few roots of unity, or a finite extension o...

We establish strong constraints on the kernel of the (reduced) Burau representation $\beta_4:B_4\to \text{GL}_3\left(\mathbb{Z}\left[q^{\pm 1}\right]\right)$ of the braid group $B_4$. We develop a theory to explicitly determine the entries of the Burau matrices of braids in $B_4$, and this is an important step toward demonstrating that $\beta_4$ is faithful (a longstanding question posed in the 1930s). The theory is based on a novel combinatorial interpretation of $\beta_4\le...

We consider the irreducible representations each of dimension 2 of the necklace braid group $\mathcal{NB}_n$ ($n=2,3,4$). We then consider the tensor product of the representations of $\mathcal{NB}_n$ ($n=2,3,4$) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $\mathcal{NB}_n$ ($n=2,3,4$) of degree 2 are unitary relative to a hermitian ...