- Volume 54 Issue 3
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.
knotted surface;marked graph diagram;singular braid monoid;surface-link
Supported by : Warsaw Center of Mathematics and Computer Science
- O. T. Dasbach, B. Gemein, A faithful representation of the singular braid monoid on three strands, in Knots in Hellas '98, 48-58, World Scientific Publishing, 2000.
- B. Gemein, Representations of the singular braid monoid and group invariants of singular knots, Topology Appl. 114 (2001), no. 2, 117-140. https://doi.org/10.1016/S0166-8641(00)00036-5
- M. Jablonowski, On a surface singular braid monoid, Topology Appl. 160 (2013), 1773-1780. https://doi.org/10.1016/j.topol.2013.07.008
- S. Kamada, Nonorientable surfaces in 4-space, Osaka J. Math. 26 (1989), no. 2, 367-385.
- A. Kawauchi, T. Shibuya, and S. Suzuki, Descriptions on surfaces in four-space. I: Normal forms, Math. Sem. Knotes Kobe Univ. 10 (1982), no. 1, 72-125.
- C. Kearton and V. Kurlin, All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron, Algebr. Geom. Topology 8 (2008), no. 3, 1223-1247. https://doi.org/10.2140/agt.2008.8.1223
- J. Kim, Y. Joung and S. Y. Lee, On generating sets of Yoshikawa moves for marked graph diagrams of surface-links, J. Knot Theory Ramications 24 (2015), no. 4, 1550018, 21 pp.
- S. J. Lomonaco, Jr., The homotopy groups of knots I. How to compute the algebraic 2-type, Pacific J. Math. 95 (1981), no. 2, 349-390. https://doi.org/10.2140/pjm.1981.95.349
- F. J. Swenton, On a calculus for 2-knots and surfaces in 4-space, J. Knot Theory Ramifications 10 (2001), no. 8, 1133-1141. https://doi.org/10.1142/S0218216501001359
- K. Yoshikawa, An enumeration of surfaces in four-space, Osaka J. Math. 31 (1994), no. 3, 497-522.
- E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 417-495. https://doi.org/10.1090/S0002-9947-1965-0202916-1