Extensions of Braid Group Representations to the Monoid of Singular Braids

Given a representation φ:B_n→G_n of the braid group B_n, n≥2 into a group G_n, we are considering the problem of whether it is possible to extend this representation to a representation Φ:SM_n→A_n, where SM_n is the singular braid monoid and A_n is an associative algebra, in which the group of units contains G_n. We also investigate the possibility of extending the representation Φ:SM_n→A_n to a representation Φ~:SB_n→A_n of the singular braid group SB_n. On the other hand, given two linear representations φ₁,φ₂:H→GL_m(k) of a group H into a general linear group over a field k, we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of SB_n which is an extension of the Lawrence–Krammer–Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence–Krammer–Bigelow representation.