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The Johnson graph $J(n, k)$ has a known genus when $k=1$, in which case it is the complete graph $K_n$. What can be said about the genus of $J(n, 2)$, or more generally $J(n, k)\ ?$

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"What can be said" is a poor way to start a question that is intended to be appropriate for MathOverflow. I suggest asking about lower bounds, or for a reference request, or something to which "Yes" or "No" will be a highly informative part of the answer. Gerhard "Ask Me About System Design" Paseman, 2012.09.09 –  Gerhard Paseman Sep 9 '12 at 18:53
    
I wonder what is the motivation behind the question. –  Felix Goldberg Sep 9 '12 at 20:53
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Bounds would be good, but I assumed that would be included in "what can be said". The question arose from music theory; certain types of scales invented by Erv Wilson with exotic names like hexanies, dekanies or eikosanies have relationships between their pitch classes in the form of Johnson graphs, and drawing the chord relationships on an orientable surface might, in theory, make such relationships more perspicuous. There are a few general classes of graphs with known formulas for the genus, so it's an interesting research problem and it seemed possible someone knew something relevant. –  Gene Ward Smith Sep 10 '12 at 21:47
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In which case a reference request is more appropriate than "tell me what I don't know" for MathOverflow. The idea is that specific and well thought out questions are likely to be answered and more helpful than broader ranging questions. If you want something appropriate for MathOverflow, include your motivation and a couple very specific questions. You may get someone who later might tackle the broader questions for you. Gerhard "Ask Me About System Design" Paseman, 2012.09.10 –  Gerhard Paseman Sep 11 '12 at 5:34
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