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Aug 17 at 18:50 comment added M. Winter Thanks Ian for your answer! Let me add to the fourth paragraph: $K_5$ and $K_{3,3}$ have exactly two non-equivalent flat/paneled embeddings. So every flat graph has a finite number of such embeddings, all of which I would consider equally "simplest".
Aug 16 at 22:44 history answered Ian Agol CC BY-SA 4.0