Algebraic intersection, lengths and Veech surfaces

TL;DR

This paper investigates the algebraic intersection properties of closed curves on translation surfaces, focusing on regular even-gons and their Teichmüller disks, extending previous studies on Veech surfaces.

Contribution

It introduces the study of the KVol invariant for regular even-gons and their Teichmüller disks, advancing understanding of curve intersections on these surfaces.

Findings

Analysis of KVol for regular even-gons with n ≥ 8

Characterization of intersection maximization on these surfaces

Extension of previous work on Veech surfaces

Abstract

In this paper, we continue the study of intersections of closed curves on translation surfaces, initiated in by S. Cheboui, A. Kessi and D. Massart for a family of arithmetic Veech surfaces and the author, E. Lanneau and D. Massart for a family of non-arithmetic Veech surfaces. Namely, we investigate the question of maximizing the algebraic intersection between two curves of given lengths by studying the quantity KVol defined for any closed orientable surface by: $$ \mathrm{KVol}(X): = \mathrm{Vol}(X,g)\cdot \sup_{\alpha,\beta} \frac{\mathrm{Int} (\alpha,\beta)}{l_g (\alpha) l_g (\beta)},$$ where the supremum is taken over all pairs of closed curves on $X$. In this paper we focus on regular $n$-gons for even $n \geq 8$ as well as their Teichm\"uller disks.