Russian Mathematical Surveys, 2023, Volume 78, Issue 2, Pages 393–395 DOI: https://doi.org/10.4213/rm10087e (Mi rm10087)

DOI: https://doi.org/10.4213/rm10087e

Funding agency	Grant number
Lomonosov Moscow State University
This research was carried out with the support of the Interdisciplinary Scientific and Educational School “Mathematical Methods for Analysis of Complex System” at Moscow State University.

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The aim of our note is to construct new representations of the (coloured) braid group using methods of parity theory and picture-valued invariants [3], [4].

Consider a braid $\beta$ on $n$ strands $\beta_k\colon[0,1]\to \mathbb R^2=\mathbb C$, $k=1,\dots,n$. Fix an index $k\in \{1,\dots,n\}$. The system of the strands $p_k(\beta)_l\colon [0,1]\to \mathbb C^*=\mathbb C\setminus\{0\}$, $l\ne k$, defined by $p_k(\beta)_l(t)=\beta_l(t)-\beta_k(t)$ forms a braid $p_k(\beta)$ on $n-1$ strands in the thickened cylinder $\mathbb C^*\times[0,1]$. The maps $z\mapsto z/|z|$ from $\mathbb C^*$ to $S^1=\{z\in\mathbb C\colon |z|=1\}$ induce a projection of the thickened cylinder $\mathbb C^*\times[0,1]$ onto the cylinder $S^1\times [0,1]$ by taking the braid $p_k(\beta)$ to a braid diagram in the latter. We also denote this diagram by $p_k(\beta)$.

In a similar way, to a pair of indices $k,l\in \{1,\dots,n\}$ there corresponds a braid $q_{k,l}(\beta)$ on $n-2$ strands,

Let $\beta'$ be a braid diagram in $S^1\times[0,1]$. Fix an arbitrary positive integer $d$. The covering map $(z,t)\mapsto (z^d,t)$ of the cylinder takes $\beta'$ to a diagram on the same number of strands. We mark as virtual crossings intersection points of strands that are not the images of crossings in $\beta'$ (Fig. 1). We denote the virtual diagram obtained by $f_d(\beta')$.

Finally, given a virtual crossing of a braid $\beta''$ in the cylinder $S^1\times[0,1]$, consider the projection of this cylinder onto a plane such that its restrictions to strands of the braid are regular and the orientation of the surface is preserved at the (classical and virtual) crossings of the diagram. On the diagram obtained by this projection we mark the images of classical crossings as classical ones, the images of virtual crossings as flat ones (marked by bullets), and the additional crossings of braids that arise after the projection as virtual crossings (Fig. 1). We denote the resulting flat-virtual braid diagram by $\psi(\beta'')$.

Zoom inZoom out

Figure 1.The maps $f_2$ and $\psi$. The cylinder is shown as a rectangle whose vertical sides are identified.

By a flat-virtual braid we mean an equivalence class of flat-virual braid diagrams modulo the isotopies of diagrams and the following moves: classical, flat, and virtual second Reidemeister moves; purely classical, flat, and virtual third Reidemeister moves; mixed third Reidemeister moves in which two crossings are flat or two crossings are virtual; far commutativity.

The flat-virtual braids on $n$ strands form a group $\operatorname{FVB}_n$ with respect to the standard operation of concatenation of diagrams. This group has the Artin generators $\sigma_{i}$, $\pi_{i}$, and $\tau_{i}$, $i=1,\dots,n-1$, corresponding to classical, flat, and virtual crossings, which are defined similarly to the case of ordinary braids.

We define the generalized Burau representation ${\widetilde \rho}$ of the flat-virtual braids $\operatorname{FVB}_n$ into the group of invertible $ n\times n $ matrices by mapping the generators $\sigma_{i}$, $\pi_{i}$, $\tau_{i}$ to block diagonal matrices with a non-trivial $ 2\times 2 $ block corresponding to the rows and columns with indices $i$ and $i+1$ that has the following form (cf. [1]):

For the proof of this theorem it is sufficient to verify that these maps agree with the Reidemeister moves.

For example, consider the element $\beta=[\psi_1{_1}\sigma_4\psi_1,\psi_2{_1}\sigma_4\sigma_3 \sigma_2\sigma_1^2\sigma_2\sigma_3\sigma_4\psi_2]\in B_5$, where $\psi_1=\sigma_3^{-1}\sigma_2\sigma_1^2\sigma_2 \sigma_4^3\sigma_3\sigma_2$ and $\psi_2=\sigma_4^{-1}\sigma_3\sigma_2 \sigma_1^{-2}\sigma_2\sigma_1^2\sigma_2^2\sigma_1\sigma_4^5$. It lies in the kernel of the Burau representation [2]. Then

Bibliography
1.	V. G. Bardakov, Sib. Èlektron. Mat. Izv., 2 (2005), 196–199
2.	S. Bigelow, Geom. Topol., 3 (1999), 397–404
3.	V. O. Manturov, Mat. Sb., 201:5 (2010), 65–110      ; English transl. in Sb. Math., 201:5 (2010), 693–733
4.	V. O. Manturov, D. Fedoseev, Seongjeong Kim, and I. Nikonov, Invariants and pictures. Low-dimensional topology and combinatorial group theory, Ser. Knots Everything, 66, World Sci. Publ., Hackensack, NJ, 2020, xxiv+357 pp.

Citation in format AMSBIB

\Bibitem{ManNik23}
\by V.~O.~Manturov, I.~M.~Nikonov
\paper Maps to virtual braids and braid representations
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 2
\pages 393--395
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\crossref{https://doi.org/10.4213/rm10087e}
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