Missing faces of neighborly and nearly neighborly polytopes and spheres

TL;DR

This paper investigates the missing face numbers of neighborly and nearly neighborly polytopes and spheres, providing bounds, characterizations, and conditions related to their combinatorial structures.

Contribution

It offers new bounds and partial characterizations of missing face numbers for neighborly and nearly neighborly polytopes and spheres, advancing understanding of their combinatorial properties.

Findings

Lower bounds on missing face numbers for certain spheres

Almost complete characterization of 2-neighborly 4-spheres

Existence of infinite families of neighborly spheres with zero missing faces

Abstract

For a $(d-1)$-dimensional simplicial complex $\Delta$ and $1\leq i\leq d$, let $f_{i-1}$ be the number of $(i-1)$-faces of $\Delta$ and $m_i$ be the number of missing $i$-faces of $\Delta$. 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 $d\geq 4$, we obtain a lower bound on $m_{\lfloor d/2\rfloor}$ in terms of $f_0$ 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.