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

TL;DR

This paper generalizes the pentagram map to higher diagonals, studying its action on spiral polygons, deriving formulas, invariants, and revealing geometric structures and orbit behaviors in the moduli space.

Contribution

It introduces the deep diagonal map $T_k$ as a generalization of the pentagram map, analyzes its action on spiral polygons, and derives new formulas and invariants for $T_3$.

Findings

$T_k$ preserves type-$eta$ and type-$eta$ $k$-spirals.

For $k=2,3$, $T_k$ has precompact orbits modulo projective transformations.

Derived a rational formula and algebraic invariants for $T_3$.

Abstract

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.