Spirals, Tic-Tac-Toe Partition, and Deep Diagonal Maps

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-$\alpha$ and type-$\beta$ $k$-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.