On biquandles for the groups $G_n^k$ and surface singular braid monoid

The groups $G_n^k$ were defined by V. O. Manturov in order to describe dynamical systems in configuration systems. In the paper we consider two applications of this theory: we define a biquandle structure on the groups $G_n^k$, and construct a homomorphism from the surface singular braid monoid to the group $G_n^2$.

[1]  Sang Youl Lee,et al.  Presentation of immersed surface-links by marked graph diagrams , 2017, Journal of Knot Theory and Its Ramifications.

[2]  Michal Jablonowski.  Presentations and representations of surface singular braid monoids , 2016, 1602.08650.

[3]  Mohamed Elhamdadi,et al.  Quandles: An Introduction to the Algebra of Knots , 2015 .

[4]  V. Manturov,et al.  On braids and groups $G_n^k$ , 2015, 1507.03745.

[5]  Jieon Kim,et al.  On generating sets of Yoshikawa moves for marked graph diagrams of surface-links , 2014, 1412.5288.

[6]  L. Kauffman.  Virtual Knot Cobordism , 2014, 1409.0324.

[7]  Michal Jablonowski.  On a surface singular braid monoid , 2012, 1211.6784.

[8]  L. Kauffman.  An Affine Index Polynomial Invariant of Virtual Knots , 2012, 1211.1601.

[9]  V. Kurlin,et al.  All 2–dimensional links in 4–space live inside a universal 3–dimensional polyhedron. , 2008, 0801.3647.

[10]  Frank Swenton.  ON A CALCULUS FOR 2-KNOTS AND SURFACES IN 4-SPACE , 2001 .

[11]  K. Yoshikawa.  An enumeration of surfaces in four-space , 1994 .

[12]  J. Birman.  New points of view in knot theory , 1993, math/9304209.

[13]  J. Baez.  Link invariants of finite type and perturbation theory , 1992, hep-th/9207041.

[14]  S. Lomonaco.  THE HOMOTOPY GROUPS OF KNOTS I. HOW TO COMPUTE THE ALGEBRAIC 2-TYPE , 1981 .

[15]  C. Rourke,et al.  The braid-permutation group , 1997 .

[16]  C. Rourke,et al.  An Introduction to Species and the Rack Space , 1993 .

[17]  T. Shibuya,et al.  Descriptions on Surfaces in Four-Space I Normal Forms , 1982 .