Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4

TL;DR

This paper constructs a counterexample demonstrating that the strong Hanani-Tutte theorem does not extend to the orientable surface of genus 4, challenging previous assumptions in topological graph theory.

Contribution

It provides the first known counterexample on genus 4 surfaces, showing limitations of the Hanani-Tutte theorem extension.

Findings

Counterexample on genus 5 graph with even crossings on genus 4 surface

Shows the strong Hanani-Tutte theorem cannot be extended to genus 4

Uses a counterexample on the torus as a base step

Abstract

We find a graph of genus $5$ and its drawing on the orientable surface of genus $4$ with every pair of independent edges crossing an even number of times. This shows that the strong Hanani-Tutte theorem cannot be extended to the orientable surface of genus $4$. As a base step in the construction we use a counterexample to an extension of the unified Hanani-Tutte theorem on the torus.