Extensions of braid group representations to the monoid of singular braids

Given a representation φ Bₙ → Gₙ of the braid group Bₙ, n ≥ 2 into a group Gₙ, we are considering the problem of whether it is possible to extend this representation to a representation Φ SMₙ → Aₙ, where SMₙ is the singular braid monoid and Aₙ is an associative algebra, in which the group of units contains Gₙ. We also investigate the possibility of extending the representation Φ SMₙ → Aₙ to a representation Φ SBₙ → Aₙ of the singular braid group SBₙ. On the other hand, given two linear representations φ₁, φ₂ H → GLₘ() of a group H into a general linear group over a field , we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of SBₙ 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.