I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?
2026-03-26 19:37:32.1774553852
Can a simple closed curve in a compact surface be dense?
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A simple closed curve in a surface $X$ is a continuous injection $f:S^1\to X$. Since $S^1$ is compact, the image of $f$ is compact and hence closed. So, the image cannot be dense (the image cannot be all of $X$ since $f$ is a homeomorphism to its image).
More generally, the same argument applies to any Hausdorff space $X$ which is not homeomorphic to $S^1$.