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arXiv · 1610.01028

Small f-vectors of 3-spheres and of 4-polytopes

Philip Brinkmann·Günter M. Ziegler
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Abstract

We present a new algorithmic approach that can be used to determine whether a given quadruple $(f_0,f_1,f_2,f_3)$ is the f-vector of any convex 4-dimensional polytope. By implementing this approach, we classify the f-vectors of 4-polytopes in the range $f_0+f_3\le22$. In particular, we thus prove that there are f-vectors of cellular 3-spheres with the intersection property that are not f-vectors of any convex 4-polytopes, thus answering a question that may be traced back to the works of Steinitz (1906/1922). In the range $f_0+f_3\le22$, there are exactly three such f-vectors with $f_0\le f_3$, namely $(10,32,33,11)$, $(10,33,35,12)$, and $(11,35,35,11)$.

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Philip Brinkmann, Günter M. Ziegler. 2016-10-04. Small f-vectors of 3-spheres and of 4-polytopes. https://arxiv.org/abs/1610.01028

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