Tiling billiards and Dynnikov's helicoid

TL;DR

This paper explores the connection between tiling billiards in cyclic quadrilaterals and the topology of plane sections of symmetric genus-3 surfaces, revealing a helicoidal construction linking these problems.

Contribution

It establishes a novel relationship between tiling billiards dynamics and Novikov's classical problem using Dynnikov's helicoid construction.

Findings

Linked tiling billiard dynamics to plane sections of genus-3 surfaces.

Proposed a helicoidal construction connecting the two problems.

Suggested potential for broader exploration in higher genus tiling billiards.

Abstract

Here are two problems. First, understand the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describe the topology of connected components of plane sections of a centrally symmetric subsurface $S \subset \mathbb{T}^3$ of genus $3$. In this note we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov's problem in higher genus seems promising, as we show in the end of this note.