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

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