Integral laminations on non-orientable surfaces

TL;DR

This paper introduces a coordinate system for integral laminations on non-orientable surfaces and establishes a bijection to a specific integer lattice, aiding in the combinatorial understanding of these structures.

Contribution

It provides explicit triangle coordinates and a bijection for integral laminations on non-orientable surfaces, extending combinatorial tools in geometric topology.

Findings

Defined triangle coordinates for integral laminations on non-orientable surfaces

Established a bijection between laminations and a lattice of integer tuples

Facilitated combinatorial analysis of laminations on complex surfaces

Abstract

We describe triangle coordinates for integral laminations on a non-orientable surface $N_{k,n}$ of genus $k$ with $n$ punctures and one boundary component, and give an explicit bijection from the set of integral laminations on $N_{k,n}$ to $(\mathbb{Z}^{2(n+k-2)}\times \mathbb{Z}^k)\setminus \left\{0\right\}$.