| Russian Mathematical Surveys |
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Brief communications Maps to virtual braids and braid representations V. O. Manturova, I. M. Nikonovb a Moscow Institute of Physics and Technology (National Research University) b Lomonosov 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,
$$
\begin{equation*}
q_{k,l}(\beta)_i(t)= \frac{\beta_i(t)-\beta_k(t)}{\beta_l(t)-\beta_k(t)}\,,\qquad i\ne k,l,\quad t\in[0,1],
\end{equation*}
\notag
$$
in the thickened cylinder $\mathbb C^*\times[0,1]$.
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 $\psi(\beta'')$. 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]):
$$
\begin{equation*}
\widetilde\rho(\sigma)=\begin{pmatrix} 1-t & t \\ 0 & 1 \end{pmatrix},\quad \widetilde\rho(\pi)=\begin{pmatrix} 0 & s \\ s^{-1} & 0 \end{pmatrix},\quad \widetilde\rho(\tau)=\begin{pmatrix} 0 & r \\ r^{-1} & 0 \end{pmatrix}.
\end{equation*}
\notag
$$
Theorem. For $k,l\in\{1,\dots,n\}$ and $d\in\mathbb N$ the compositions and are well-defined representations of the braid group $B_n$. 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
$$
\begin{equation*}
\widetilde \rho\circ\psi\circ f_2\circ p_1(\beta)|_{t=-1, s=1, r=1}= \begin{pmatrix} 481 & -880 & 800 & -400 \\ 480 & -879 & 800 & -400 \\ 480 & -880 & 801 & -400\\ 480 & -880 & 800 & -399 \end{pmatrix} \ne 1.
\end{equation*}
\notag
$$
Thus, the family of representations $\widetilde \rho\circ\psi\circ f_d\circ p_k$ is a stronger braid invariant than the Burau representation (note that we obtain the standard Burau representation by adding an additional $(n+1)$st free strand and considering the image of the representation $\widetilde\rho\circ\psi\circ f_1\circ p_{n+1}$).
Citation in format AMSBIB
\Bibitem{ManNik23} |
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