# Unsensed enumeration of cubic unicellular maps on orientable and non-orientable surfaces

## TL;DR

This paper develops a method to count cubic unicellular maps on both orientable and non-orientable surfaces without considering symmetries, providing explicit formulas and numerical data.

## Contribution

It introduces an orbifold-based approach to enumerate unsensed cubic unicellular maps, connecting them to quotient and rooted maps on simpler surfaces.

## Key findings

- Explicit formulas for orientable surfaces using known sensed counts.
- Finite sum expressions for non-orientable surfaces.
- Numerical tables and asymptotic insights included.

## Abstract

We enumerate cubic (3-regular) unicellular maps on closed surfaces up to all homeomorphisms. Using the orbifold approach, we reduce the unsensed enumeration to explicit counts of quotient maps and rooted cubic/precubic maps on simpler surfaces. For orientable hosts this yields a compact identity expressed through known sensed and rooted numbers; for non orientable hosts we obtain a fully explicit finite sum expression via precubic counts. Numerical tables are provided, together with a brief asymptotic discussion.