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Compute the Poincaré polynomial for a simplicial complex
ResourceFunction["PoincarePolynomial"][cplx] gives the Poincaré polynomial for the specified simplicial complex cplx. |
simplex | a simplex as defined in the table below |
{simplex1,simplex2,…} | a list of simplices |
{{v11,v12,…},…} | a list of lists, where the vi,j correspond to simplex vertices |
Graph[…] | a graph object |
MeshRegion[…] | a mesh region |
Get the Poincaré polynomial for a simplex:
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Get the Poincaré polynomial for a simplicial complex:
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Get the Poincaré polynomial for a MeshRegion:
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Specify the variable:
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Obtain the Poincaré polynomial as a Function:
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Obtain a Series:
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Many graphics primitives that represent simplices can be used:
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A simplicial complex can be specified as lists of indices:
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Get the Poincaré polynomial for a graph:
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The coefficient of xk in the Poincaré polynomial corresponds to the Betti number bk:
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PoincarePolynomial only considers unique vertices in each simplex:
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As graphs, these are considered distinct:
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The same is true for hypergraphs:
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All graphs are treated as undirected:
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This means that the first coefficient corresponds to the number of WeaklyConnectedGraphComponents instead of ConnectedGraphComponents:
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The Poincaré polynomial does not identify links, knots or braids:
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The Poincaré polynomial cannot determine orientability:
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PoincarePolynomial only allows Polygon objects with three vertices:
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The Poincaré–Birdnardo polynomial is a special case of the Poincaré polynomial:
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