A marked graph diagram of a knotted surface is a 4-regular spatial graph with a marker at each 4-valent vertex, which presents a knotted surface. Two marked graph diagrams presenting the same type of knotted surface are related by a finite sequence of local moves called Yoshikawa moves and an isotopy of the diagram in \(\mathbb{R}^2\); see [K. Yoshikawa, Osaka J. Math. 31, No. 3, 497–522 (1994; Zbl 0861.57033); F. J. Swenton, J. Knot Theory Ramifications 10, No. 8, 1133–1141 (2001; Zbl 1001.57044)]. By presenting a marked graph diagram by a form of the geometric closure of a singular braid of \(n\) strands with a marker at each singular crossing, we obtain a monoid \(SSB_n\), called the surface singular braid monoid.
The main results are as follows. No representation \(\rho: SSB_n \to M_2(K)\) for \(n \geq 2\) is faithful, and no representation \(\rho: SSB_n \to M_3(K)\) for \(n \geq 3\) is faithful, where \(M_m(K)\) is the multiplicative monoid of \(m \times m\) matrices with entries valued in a field \(K\). Further, the author gives new presentations for \(SSB_n\) other than the one given by the author in [Topology Appl. 160, No. 13, 1773–1780 (2013; Zbl 1298.57019)], and gives in Table 1 in the paper the surface singular braid formulations of the knotted surfaces in Yoshikawa’s table.