Connection points on double regular polygons

TL;DR

This paper investigates connection points on double regular n-gon translation surfaces, identifying families of points with specific properties and conjecturing about their connection status, especially for n=7.

Contribution

It introduces a large family of non-connection points with trace field coordinates and provides a constructive proof for prime n, advancing understanding of the surface's connection structure.

Findings

Identifies non-connection points with trace field coordinates for n ≠ 9

Conjectures all remaining points are connection points for n=7

Constructs explicit separatrix for prime n, showing non-extension to saddle connection

Abstract

We study connection points on the double regular $n$-gon translation surface, for $n \geq 7$ odd and its staircase model. For $n \neq 9$, we provide a large family of points with coordinates in the trace field that are not connection points. This family includes the central points, and for $n=7$ we conjecture that all the remaining points are connection points. Further, in the case where $n \geq 7$ is a prime number, we provide a constructive proof by exhibiting an explicit separatrix passing through a central point that does not extend to a saddle connection.