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Vertex maps between \(\triangle \), \(\Box \), and \(\Diamond \)

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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

  1. In [3] the vertex set is denoted by \(\mathrm{vert }(\mathrm{Hom }(P,Q))\).

References

  1. Akopyan, A., Karasev, R.: Inscribing a regular octahedron into polytopes. Discrete Math. 313, 122–128 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Billera, L.J., Sturmfels, B.: Fiber polytopes. Ann. Math. (2) 135(3), 527–549 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bogart, T., Contois, M., Gubeladze, J.: Hom-polytopes. Math. Z 273, 1267–1296 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bruns, W., Gubeladze, J.: Polytopes, Rings, and \(K\)-Theory. Springer Monographs in Mathematics. Springer, Dordrecht (2009)

  5. Coxeter, H.S.M.: Regular Polytopes, 2nd edn. Dover Publications Inc., New York (1973)

    Google Scholar 

  6. Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Polytopes-combinatorics and computation (Oberwolfach, 1997), volume 29 DMV Seminar, pp. 43–73. Birkhäuser, Basel (2000)

  7. Riordan, J.: Combinatorial Identities. Wiley, New York (1968)

    MATH  Google Scholar 

  8. Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Published electronically at http://oeis.org (2008)

  9. Valby, L.: A category of polytopes. Available at http://people.reed.edu/davidp/homepage/students/valby.pdf

  10. Ziegler, G.: Lectures on polytopes, volume 152 of Graduate Texts in Mathematics, revised edn. Springer, New York (1998)

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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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Correspondence to Joseph Gubeladze.

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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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