In this post I am going to describe some of the experiments I have been running to understand the combinatorial properties of polytopal graphs.
Having created a polytope enumeration system we now have access to millions of 6d, 7d, 8d and even higher dimensional polytopes. The information about the polytopes consists of their face lattice only (the combinatorial information, not their geometrical realization in d-space). The combinatorial face lattice itself has interesting problems associated with it; the fundamental being a full characterization of the allowable f-vectors of a given (n,d)-polytope. But even before that we can address several low hanging fruit once experimental data has been generated. Consider for example the (4,11) polytope data set we have. As of today we have generated close to a million combinatorically unique polytopes.
As described in my earlier post we have implemented polytope enumeration software which starts off with the n-dimensional simplex and finds vertex sets to be cut-off resulting in a higher facet count polytope.
We hypothesize that the path from the simplex to any given polytope can be uniquely identified, although multiple paths exist (atleast n paths exist, where n is the number of facets of the created polytope). We order these paths using vertex labels. The location of any given polytope in this infinite tree can be identified by the cut set vertices from the simplex. Alternatively I am hypothesizing that the betweeness centrality index of the cut set also determines the height (how many other polytopes can be generated from a given polytope). This hypothesis will be experimentally verified in this post.
Having created a polytope enumeration system we now have access to millions of 6d, 7d, 8d and even higher dimensional polytopes. The information about the polytopes consists of their face lattice only (the combinatorial information, not their geometrical realization in d-space). The combinatorial face lattice itself has interesting problems associated with it; the fundamental being a full characterization of the allowable f-vectors of a given (n,d)-polytope. But even before that we can address several low hanging fruit once experimental data has been generated. Consider for example the (4,11) polytope data set we have. As of today we have generated close to a million combinatorically unique polytopes.
As described in my earlier post we have implemented polytope enumeration software which starts off with the n-dimensional simplex and finds vertex sets to be cut-off resulting in a higher facet count polytope.
We hypothesize that the path from the simplex to any given polytope can be uniquely identified, although multiple paths exist (atleast n paths exist, where n is the number of facets of the created polytope). We order these paths using vertex labels. The location of any given polytope in this infinite tree can be identified by the cut set vertices from the simplex. Alternatively I am hypothesizing that the betweeness centrality index of the cut set also determines the height (how many other polytopes can be generated from a given polytope). This hypothesis will be experimentally verified in this post.
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