April 8th, 2018, 11:04 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory  Genus given from pole minima
Can the minimum number of singularities on a three dimensional surface within continuous lines (exhaustively mapped over a topological space) help describe the value of its genus? For instance, think of a globe with two separate poles (singularities) which can be located anywhere on the sphere. The malleable continuous lines may be stretched to accommodate the two poles, but no fewer. This is an example of genuseven. No poles might be needed for genusodd shapes as their holes may interchange topologically like their genusone poles. Each pair of poles (even) corresponds to and transforms into a hole (odd). When forms of differing genera meld, do they conserve the topological properties of poles and holes? So, do genera = f(poles)? 

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genus, minima, pole 
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