On a surface singular braid monoid

摘要

We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type. Then we state relations on words that follow from topological Yoshikawa moves. As a direct application we will reprove some known theorem about twist-spun knots. We wish then to investigate an index associated to the closure of surface singular braid. Using our relations we will prove that there are exactly six types of knotted surfaces with the index less or equal to two, and there are infinitely many types of surface-knots with index equal to three. Towards the end we will construct a family of classical diagrams such that to unlink them requires at least four Reidemeister Ⅲ moves.