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.

中文翻译：

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Hanani-Tutte 定理在属 4 表面上的推广的反例

我们找到了 $5$ 属的图及其在 $4$ 属的可定向曲面上的图，其中每对独立边交叉偶数次。这表明强 Hanani-Tutte 定理不能推广到 $4$ 属的可定向曲面。作为构造的基本步骤，我们使用一个反例来扩展环面上的统一 Hanani-Tutte 定理。