Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs
Let G be a connected bipartite graph with colour classes E and V and root polytope Q. Regarding the hypergraph H=(V,E) induced by G, we prove that the interior polynomial of H is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It foll...
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| Format: | Article |
| Language: | English |
| Published: |
Oxford University Press (OUP),
2018-05-30T18:48:50Z.
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| Subjects: | |
| Online Access: | Get fulltext |
| Summary: | Let G be a connected bipartite graph with colour classes E and V and root polytope Q. Regarding the hypergraph H=(V,E) induced by G, we prove that the interior polynomial of H is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of H and its transpose H=(E,V) agree. When G is a complete bipartite graph, our result recovers a well-known hypergeometric identity due to Saalschütz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read off of an associated Floer homology group. National Science Foundation (U.S.) (Grant DMS‐1100147) National Science Foundation (U.S.) (Grant DMS‐1362336) |
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