On the monoid of singular braids

We show that the singular braid monoid on 3 strands embeds in the integral group ring of the braid group proving a special case of a conjecture originating from J.S. Birman (1993). The statement is proven by developing general results connected to the conjecture, including an extension of Artinʹs braiding theorem to a special class of singular braids. We prove two corollaries which are also in strong connection with the conjecture and discuss the difficulties arising when the number of strands is greater than 3.