2024/12/20 by Zheng‐Yu Zou, Zou, Zhengyu · 2 citations
Computer Science · #05E99 (Secondary) #37J70 (Primary) 51A05 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2412.15561
openalex publication_date 2024/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The deep diagonal map Tk acts on planar polygons by connecting the k-th diagonals and intersecting them successively. The map T2 is the pentagram map, and Tk is a generalization. We study the action of Tk on two subsets of the so-called twisted polygons, which we term type-α and type-β k-spirals. For k ≥ 2, Tk 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 Tk-orbits modulo projective transformations. We derive a rational formula for T3, 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 T3. These special orbits in the moduli space are partitioned into cells of a 3 × 3 tic-tac-toe grid. This establishes the action of Tk on k-spirals as a geometric generalization of T2 on convex polygons.