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

TL;DR

This paper introduces a new algorithm to determine if a quadruple of numbers can be the f-vector of a convex 4-polytope, classifies such f-vectors up to a certain size, and shows some f-vectors correspond to cellular 3-spheres but not to convex 4-polytopes.

Contribution

The paper develops an algorithmic method to classify f-vectors of 4-polytopes and identifies specific f-vectors that cannot be realized as convex 4-polytopes, answering longstanding questions.

Findings

Classified f-vectors of 4-polytopes with $f_0+f_3 \,\leq 22$

Proved existence of cellular 3-spheres with certain f-vectors not realizable as convex 4-polytopes

Identified exactly three such non-realizable f-vectors within the range

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 $(11,35,35,11)$.