| Russian Mathematical Surveys |
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Brief communications Maps from knots in the cylinder to flat-virtual knots V. O. Manturovab, I. M. Nikonovc a Moscow Institute of Physics and Technology b Nosov Magnitogorsk State Technical University c Lomonosov Moscow State University
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     The aim of this paper is to introduce a new invariant of knots in the thickened cylinder (torus) which takes values in flat-virtual knots. Given the cylinder $C=S^{1}\times [0,1]$ with the angular coordinate $\alpha\in [0,2\pi=0)$ and the vertical coordinate $z\in [0,1]$, fix a natural number $d$ and say that two points $A=(\alpha_1,z_1)$ and $B=(\alpha_2,z_2)$ of the cylinder are equivalent if $z_1=z_2$ and $\alpha_1-\alpha_2=2\pi k/d$ for an integer $k$. Similarly, in the torus $T^{2}$ with angular coordinates $(\alpha,\beta)\in [0,2\pi=0)^2$ let us fix a lattice (a discrete subgroup) $l$. For example, $l=\{(2\pi k_{1}/d_{1},2\pi k_{2}/d_{2})\mid k_1,k_2\in\mathbb Z\}$ for some $d_{1},d_{2}\in\mathbb N$. We call points $A,B\in T^2$ equivalent if $A-B\in l$. Let $K$ be a link diagram in the cylinder (torus) and $d$ be an integer ($l$ be a lattice, respectively). Assume that the diagram $K$ is in general position with respect to the lattice, that is, two arbitrary different equivalent points $e,e'\in K$ are not crossings, and the tangent vectors of the diagram at $e$ and $e'$ are not collinear. We construct a new diagram $\phi_{d}(K)$ (respectively, $\phi_{l}(K)$) with classical, flat, and virtual crossings as follows. In both cases ($\phi$ or $\phi'$), for each pair of equivalent points on the edges, say $(e_{j},e'_{j})$, we create a new flat crossing. Let $e'_{j}= e_{j}+g$, where $g$ is an element of the corresponding group ($\mathbb{Z}_{d}$ or $l$). We choose one of the equivalent points, say $e_{j}$, remove a small neighbourhood $N=U(e_{j})$ with endpoints $A$ and $B$, shift the arc $AB$ by $g$, connect $A$ with $A+g$ and $B$ with $B+g$ by arbitrary curves not passing through crossings, and mark the intersections in the curves $[A,A+g]$, $[B,B+g]$ by virtual crossings (see Fig. 1, (a)). Definition 1. A flat-virtual link is an equivalence class of flat-virtual diagrams modulo the following moves: (1) classical Reidemeister moves on classical crossings (there are three such moves in the non-orientable case); (2) flat second and flat third Reidemeister moves (these moves are obtained from the classical ones by forgetting the information about overpasses and underpasses); (3) mixed flat- classical third Reidemeister moves when a strand with two consecutive flat crossings passes through a classical crossing (the type of the classical crossing is preserved); (4) virtual detour moves: a strand containing only virtual crossings and self- crossings can be removed and replaced by a strand with the same endpoints where all new intersections are marked by virtual crossings. By removing the flat third Reidemeister move from the list of moves we obtain the definition of a restricted flat-virtual link. Theorem 1. The maps $\phi_{d}$ (or $\phi_{l}'$) are well-defined maps from links on the thickened cylinder (respectively, thickened torus) to flat-virtual links. In other words, if $K$ and $K'$ are isotopic diagrams on the cylinder (torus), then $\phi_{d}(K)\sim \phi_{d}(K')$ (respectively, $\phi'_{l}(K)\sim \phi'_{l}(K)$), where $\sim$ denotes the equivalence of diagram with respect to the moves defined above. Moreover, in the case $d=2$ the diagrams $\phi_{2}(K)$ and $\phi_{2}(K')$ define the same strict flat-virtual link. To illustrate this theorem consider the Whitehead link (Fig. 1, (b), top left). By removing the trivial (gray) component we obtain a knot in the full torus (Fig. 1, (b), top right). For $d=2$ we have two pairs of equivalent points ($(A,A)$ and $(B,B)$; Fig. 1, (b), bottom right). Applying $\phi$ (and forgetting the structure of the full torus) we obtain a flat-virtual diagram $W$ at the bottom left. To show the non-triviality of the diagram we introduce a flat-virtual Jones polynomial. Let $D$ be a flat-virtual link diagram and $C(D)$ be the set of its classical crossings. A state $s\in\{0,1\}^{C(D)}$ specifies a smoothing $D_s$ of the diagram $D$ which is obtained by applying the following smoothing rule to each crossing in $D$:  Set $\alpha(s)=|s^{-1}(0)|$ and $\beta(s)=|s^{-1}(1)|$, and let $\gamma_{\rm even}(s)$ (respectively, $\gamma_{\rm odd}(s)$) be the number of components in $D_s$ which have an even (respectively, odd) number of flat crossings, and $w(D)$ be the writhe number of the diagram $D$. We define the flat-virtual Jones polynomial $X(D)\in\mathbb{Z}[a,a^{-1},b]$ of the diagram $D$ by the formula (cf. [1])
$$
\begin{equation*}
X(D)=(-a)^{-3w(D)}\sum_{s\in \{0,1\}^{C(D)}}a^{\alpha(s)- \beta(s)}(-a^2-a^{-2})^{\gamma_{\rm even}(s)}b^{\gamma_{\rm odd}(s)}.
\end{equation*}
\notag
$$
For the diagram $W$ we have $X(W)=(a^{-6}-a^{-2})b^2-a^{-10}-a^{-6}$; thus, the flat-virtual link $W$ is non-trivial. In the future we aim to improve the polynomial $X$ to a picture-valued invariant (see [2] and [3]). 
Citation in format AMSBIB
\Bibitem{ManNik24} |
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