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Representations of Polytopes
Dobbins, Michael Gene
Dobbins, Michael Gene
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2011
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Mathematics
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http://dx.doi.org/10.34944/dspace/1091
Abstract
Here we investigate a variety of ways to represent polytopes and related objects. We define a class of posets, which includes all abstract polytopes, giving a unique representative among posets having a particular labeled flag graph and characterize the labeled flag graphs of abstract polytopes. We show that determining the realizability of an abstract polytope is equivalent to solving a low rank matrix completion problem. For any given polytope, we provide a new construction for the known result that there is a combinatorial polytope with a specified ridge that is always projectively equivalent to the given polytope, and we show how this makes a naturally arising subclass of intractable problems tractable. We give necessary and sufficient conditions for realizing a polytope's interval poset, which is the polytopal analog of a poset's Hasse diagram. We then provide a counter example to the general realizablity of a polytope's interval poset.
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