Abstract
We study the vertices of the polytopes of all affine maps (a.k.a. hom-polytopes) between higher dimensional simplices, cubes, and crosspolytopes. Systematic study of general hom-polytopes was initiated in [3]. The study of such vertices is the classical aspect of a conjectural homological theory of convex polytopes. One quickly runs into open problems even for simple source and target polytopes. The vertices of \(\mathrm{Hom }(\triangle _m,-)\) and \(\mathrm{Hom }(-,\Box _n)\) are easily understood. In this work we describe the vertex sets of \(\mathrm{Hom }(\Box _m,\triangle _n)\), \(\mathrm{Hom }(\Diamond _m,\triangle _n)\), and \(\mathrm{Hom }(\Diamond _m,\Diamond _n)\). The emergent pattern in our arguments is reminiscent of diagram chasing in homological algebra.
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Notes
In [3] the vertex set is denoted by \(\mathrm{vert }(\mathrm{Hom }(P,Q))\).
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Acknowledgments
We thank Brian Cruz for computing \(\beta (5)\) and the anonymous referee whose comments helped improving the paper.
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Joseph Gubeladze was supported by NSF Grants DMS-1000641 and DMS-1301487.
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Gubeladze, J., Love, J. Vertex maps between \(\triangle \), \(\Box \), and \(\Diamond \) . Geom Dedicata 176, 375–399 (2015). https://doi.org/10.1007/s10711-014-9973-3
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DOI: https://doi.org/10.1007/s10711-014-9973-3