Spirals, Tic-Tac-Toe Partition, and Deep Diagonal Maps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911170176221184 |
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| author | Zou, Zhengyu |
| author_facet | Zou, Zhengyu |
| contents | The deep diagonal map $T_k$ acts on planar polygons by connecting the $k$-th diagonals and intersecting them successively. The map $T_2$ is the pentagram map, and $T_k$ is a generalization. We study the action of $T_k$ on two subsets of the so-called twisted polygons, which we term type-$α$ and type-$β$ $k$-spirals. For $k \geq 2$, $T_{k}$ preserves both types of $k$-spirals. In particular, we show that for $k = 2$ and $k = 3$, both types of $k$-spirals have precompact forward and backward $T_k$-orbits modulo projective transformations. We derive a rational formula for $T_3$, which generalizes the $y$-variables transformation formula of the corresponding quiver mutation by M. Glick and P. Pylyavskyy. We also present four algebraic invariants of $T_3$. These special orbits in the moduli space are partitioned into cells of a $3 \times 3$ tic-tac-toe grid. This establishes the action of $T_k$ on $k$-spirals as a geometric generalization of $T_2$ on convex polygons. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_15561 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spirals, Tic-Tac-Toe Partition, and Deep Diagonal Maps Zou, Zhengyu Dynamical Systems Combinatorics 37J70 (Primary) 51A05, 05E99 (Secondary) The deep diagonal map $T_k$ acts on planar polygons by connecting the $k$-th diagonals and intersecting them successively. The map $T_2$ is the pentagram map, and $T_k$ is a generalization. We study the action of $T_k$ on two subsets of the so-called twisted polygons, which we term type-$α$ and type-$β$ $k$-spirals. For $k \geq 2$, $T_{k}$ preserves both types of $k$-spirals. In particular, we show that for $k = 2$ and $k = 3$, both types of $k$-spirals have precompact forward and backward $T_k$-orbits modulo projective transformations. We derive a rational formula for $T_3$, which generalizes the $y$-variables transformation formula of the corresponding quiver mutation by M. Glick and P. Pylyavskyy. We also present four algebraic invariants of $T_3$. These special orbits in the moduli space are partitioned into cells of a $3 \times 3$ tic-tac-toe grid. This establishes the action of $T_k$ on $k$-spirals as a geometric generalization of $T_2$ on convex polygons. |
| title | Spirals, Tic-Tac-Toe Partition, and Deep Diagonal Maps |
| topic | Dynamical Systems Combinatorics 37J70 (Primary) 51A05, 05E99 (Secondary) |
| url | https://arxiv.org/abs/2412.15561 |