Abstract
Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is a fundamental problem in phylogenetic theory. We show that BME polytope has a sublattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sublattice corresponds to compatible sets of splits displayed by phylogenetic trees and extends the lattice of faces of the BME polytope found by Hodge, Haws and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Nearly all of these turn out to actually be facets of the BME polytope, a collection of facets which grows exponentially.
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Acknowledgements
We thank the editors and both referees for helpful comments. Stefan Forcey would like to thank the organizers and participants in the working group for geometric approaches to phylogenetic tree reconstructions, at the NSF/CBMS Conference on Mathematical Phylogeny held at Winthrop University in June–July 2014. Especially helpful were conversations with Ruriko Yoshida, Terrell Hodge and Matt Macauley. Stefan Forcey would also like to thank the American Mathematical Society and the Mathematical Sciences Program of the National Security Agency for supporting this research through Grant H98230-14-0121. (This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints.) Stefan Forcey’s specific position on the NSA is published in Forcey (2014). Suffice it to say here that he appreciates NSA funding for open research and education, but encourages reformers of the NSA who are working to ensure that protections of civil liberties keep pace with intelligence capabilities.
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Forcey, S., Keefe, L. & Sands, W. Split-Facets for Balanced Minimal Evolution Polytopes and the Permutoassociahedron. Bull Math Biol 79, 975–994 (2017). https://doi.org/10.1007/s11538-017-0264-7
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DOI: https://doi.org/10.1007/s11538-017-0264-7