Missing faces of neighborly and nearly neighborly polytopes and spheres

2025/05/27 by Novik, Isabella, Zheng, Hailun
#05E45 #52B05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.20699

Abstract

For a (d-1)-dimensional simplicial complex Δ and 1≤ i≤ d, let f_i-1 be the number of (i-1)-faces of Δ and m_i be the number of missing i-faces of Δ. In the nineties, Kalai asked for a characterization of the m-numbers of simplicial polytopes and spheres -- a problem that remains wide open to this day. Here, we study the m-numbers of nearly neighborly and neighborly polytopes and spheres. Specifically, for d≥ 4, we obtain a lower bound on m_{\lfloor d/2\rfloor} in terms of f₀ and f_{\lfloor d/2\rfloor-1} in the class of all (\lfloor d/2\rfloor-1)-neighborly (d-1)-spheres. For neighborly spheres, we (almost) characterize the m-numbers of 2-neighborly 4-spheres, and we show that, for all odd values of k, there exists an infinite family of neighborly simplicial 2k-spheres with m_k+1=0. Along the way, we provide a simple numerical condition based on the m-numbers that allows to establish non-polytopality of some neighborly odd-dimensional spheres.

Citations

• Affine stresses, inverse systems, and reconstruction problems
• On the anisotropy theorem of Papadakis and Petrotou
• Many neighborly spheres
• Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles
• Positive Plücker tree certificates for non-realizability
• The characteristic 2 anisotropicity of simplicial spheres
• New families of highly neighborly centrally symmetric spheres
• Combining realization space models of polytopes
• Combinatorial Lefschetz theorems beyond positivity
• Ear Decomposition and Balanced Neighborly Simplicial Manifolds
• The flag upper bound theorem for 3- and 5-manifolds
• On the generalized lower bound conjecture for polytopes and spheres
• Many neighborly polytopes and oriented matroids
• The number of cliques in graphs of given order and size
• Empty simplices of polytopes and graded Betti numbers
• Many Triangulated 3-Spheres

Related

• Remarks on missing faces and generalized lower bounds on face numbers
• A non-face characterization of spheres on few vertices
• Balanced complexes and complexes without large missing faces
• Many neighborly spheres
• Cubulating the sphere with many facets