Extensions of braid group representations to the monoid of singular braids

Abstract

Given a representation $φ\colon B_n \to G_n$ of the braid group $B_n$, $n \geq 2$ into a group $G_n$, we are considering the problem of whether it is possible to extend this representation to a representation $Φ\colon SM_n \to A_n$, 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 $Φ\colon SM_n \to A_n$ to a representation $\widetildeΦ \colon SB_n \to A_n$ of the singular braid group $SB_n$. On the other hand, given two linear representations $φ_1, φ_2 \colon H \to GL_m(\Bbbk)$ of a group $H$ into a general linear group over a field $\Bbbk$, 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.