Journal of the
Korean Mathematical Society JKMS

The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful. We will also derive new presentations for the surface singular braid monoid, one with reduced the number of defining relations, and the other with reduced the number of its singular generators. We include surface singular braid formulations of all knotted surfaces in Yoshikawa's table.

Keywords: knotted surface, marked graph diagram, singular braid monoid, surface-link

MSC numbers: Primary 57Q45; Secondary 20M30