Reconstructing simplicial polytopes from their graphs and affine $2$-stresses

TL;DR

This paper proves a special case of Kalai's 1994 conjecture, showing that the combinatorial structure of a simplicial polytope can be reconstructed from its graph and affine 2-stresses, advancing understanding of polytope reconstruction.

Contribution

The paper establishes the first non-trivial case of Kalai's conjecture (k=2) and confirms the conjecture for k-neighborly polytopes, providing new insights into polytope reconstruction.

Findings

Proved the case of k=2 in Kalai's conjecture.

Confirmed Kalai's conjecture for k-neighborly polytopes.

Demonstrated that the combinatorial type is determined by the graph and affine 2-stresses.

Abstract

A conjecture of Kalai from 1994 posits that for an arbitrary $2\leq k\leq \lfloor d/2 \rfloor$, the combinatorial type of a simplicial $d$-polytope $P$ is uniquely determined by the $(k-1)$-skeleton of $P$ (given as an abstract simplicial complex) together with the space of affine $k$-stresses on $P$. We establish the first non-trivial case of this conjecture, namely, the case of $k=2$. We also prove that for a general $k$, Kalai's conjecture holds for the class of $k$-neighborly polytopes.