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With Alexander Metzger I co-authored this paper about an algorithm which, for an arbitrary graph (G) with (n) vertices, this algorithm determines the orientable genus in (\mathcal{O}\left(\frac{2^{n^2+3n}}{n^{n+1}}\right)) steps. You can access the paper here.
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This is a paper I wrote describing the Maximum Cut problem and the Goemanns-Williamson algorithm. Give it a read!