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The Kuratowski graphs $K_{3,3}$ and $K_5$ characterize planarity. Countingdistinct 2-cell embeddings of these two graphs on orientable surfaces waspreviously done by using Burnside's Lemma and their automorphism groups,without actually constructing the embeddings. We obtain all 2-cell embeddingsof these graphs on the double torus, using a constructive approach. This showsthat there is a unique non-orientable 2-cell embedding of $K_{3,3}$, 14orientable and 17 non-orientable 2-cell embeddings of $K_5$ on the doubletorus, which explicitly confirms the enumerative results. As a consequence,several new polygonal representations of the double torus are presented.

Embedding K3,3 and K5 on the Double Torus