Geodesic
The Groups~$G_{n}^{2}$ with Additional Structures
Matematičeskie zametki, Tome 103 (2018) no. 4, pp. 549-567.

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In the paper [1], V. O. Manturov introduced the groups $G_{n}^{k}$ depending on two natural parameters $n>k$ and naturally related to topology and to the theory of dynamical systems. The group $G_{n}^{2}$, which is the simplest part of $G_{n}^{k}$, is isomorphic to the group of pure free braids on $n$ strands. In the present paper, we study the groups $G_{n}^{2}$ supplied with additional structures – parity and points; these groups are denoted by $G_{n,p}^{2}$ and $G_{n,d}^{2}$. First, we define the groups $G_{n,p}^{2}$ and $G_{n,d}^{2}$, then study the relationship between the groups $G_{n}^{2}$, $G_{n,p}^{2}$, and $G_{n,d}^{2}$. Finally, we give an example of a braid on $n+1$ strands, which is not the trivial braid on $n+1$ strands, by using a braid on $n$ strands with parity. After that, the author discusses links in $S_{g} \times S^{1}$ that can determine diagrams with points; these points correspond to the factor $S^{1}$ in the product $S_{g} \times S^{1}$.
Keywords: braids, free braids, knots, links, parity, braid groups with parity. 
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Kim Seongjeong. The Groups~$G_{n}^{2}$ with Additional Structures. Matematičeskie zametki, Tome 103 (2018) no. 4, pp. 549-567. http://geodesic.mathdoc.fr/item/MZM_2018_103_4_a6/

[1] V. O. Manturov, Non-Reidemeister Knot Theory and its Applications in Dynamical Systems, Geometry, and Topology, 2015, arXiv: 1501.05208v1

[2] V. O. Manturov, I. M. Nikonov, “On braids and groups $G_{n}^{k}$”, J. Knot Theory Ramifications, 24:13 (2015), 1541009 | MR | Zbl

[3] V. G. Bardakov, “The virtual and universal braids”, Fund. Math., 184 (2004), 1–18 | DOI | MR | Zbl

[4] V. G. Bardakov, P. Bellingeri, C. Damianti, “Unrestricted virtual braids, fused links and other quotients of virtual braid group”, J. Knot Theory Ramifications, 24:12 (2015), 1550063 | DOI | MR | Zbl

[5] S. Kim, V. O. Manturov, “The group $G_{n}^{2}$ and Invariants of free links valued in free groups”, J. Knot Theory Ramifications, 24:13 (2015), 1541010 | MR | Zbl

[6] D. A. Fedoseev, V. O. Manturov, “On marked braid groups”, J. Knot Theory Ramifications, 24:13 (2015), 1541005 | DOI | MR | Zbl

[7] M. Goussarov, M. Polyak, O. Viro, “Finite-type invariants of classical and virtual knots”, Topology, 39:5 (Sep 2000), 1045–1068 | DOI | MR | Zbl

[8] R. Fenn, R. Rimanyi, C. Rourke, “The braid-permutation group”, Topology, 36:1 (1997), 123–135 | DOI | MR | Zbl

[9] V. O. Manturov, On Groups $G_{n}^{2}$ and Coxeter Goups, 2015, arXiv: 1512.09273v1

[10] V. O. Manturov, “Chetnost v teorii uzlov”, Matem. sb., 201:5 (2010), 65–110 | DOI | MR | Zbl

[11] V. O. Manturov, D. P. Ilyutko, Virtual Knots. The State of the Art, World Sci., Singapore, 2013 | MR | Zbl