What is the number of decompositions of torus into given number of regions by unions of geodesics?

TL;DR

This paper investigates the combinatorial problem of counting the number of ways to decompose a 2D torus into a specified number of regions using unions of geodesics, and explores possible region counts for circle arrangements on the plane.

Contribution

It provides preliminary results addressing two open questions about decompositions of the torus and circle arrangements, advancing understanding of geometric partitioning problems.

Findings

Derived bounds for the number of regions in torus decompositions

Characterized possible region counts for circle arrangements with pairwise intersections

Extended known results on geometric decompositions and arrangements

Abstract

We prove some preliminary results concerning two questions of O.Karpenkov: (1) What is the number of decompositions of torus of dimension 2 into given number f of regions by unions of n geodesics? (2) On the plane there are n circles not in general position, every pair of cicles has at least one common point. What is the set of all possible numbers of regions?