Extensions of Braid Group Representations to the Monoid of Singular Braids

Given a representation \(\varphi :B_n \rightarrow G_n\) of the braid group \(B_n\), \(n \ge 2\) into a group \(G_n\), we are considering the problem of whether it is possible to extend this representation to a representation \(\Phi :SM_n \rightarrow A_n\), 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 \(\Phi :SM_n \rightarrow A_n\) to a representation \(\widetilde{\Phi } :SB_n \rightarrow A_n\) of the singular braid group \(SB_n\). On the other hand, given two linear representations \(\varphi _1, \varphi _2 :H \rightarrow GL_m(\Bbbk )\) 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.