Representations of the singular braid monoid and group invariants of singular knots

It is well known that the Artin representation of the braid group may be used to calculate the fundamental group of knot complements while the Burau representation can be used to calculate the Alexander polynomial of knots. In this note we will study extensions of the Artin representation and the Burau representation to the singular braid monoid and the relation between them which are induced by Foxʹ free calculus. g singular braids, we obtain singular knots as they appear in the theory of Vassiliev invariants. Thus the extensions of the Artin representation and the Burau presentation give rise to invariants of singular knots. Here, we will focus on the invariants coming from the extended Artin representations. We obtain an infinite family of group invariants, all of them in relation with the fundamental group of the knot complement.