부산대학교 도서관

Kalai conjectured that if P is a simplicial d-polytope that has no missing faces of dimension d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1), then the graph of P and the space of affine 2-stresses of P determine P up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1) and P is a simplicial d-polytope that has no missing faces of dimension d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1), then the space of affine i-stresses of P determines the space of affine 1-stresses of P. We prove this conjecture for (1) k-stacked d-polytopes with d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1), (2) d-polytopes that have no missing faces of dimension d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1), and (3) flag PL d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1)-spheres with generic embeddings (for all d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1)). We also discuss several related results and conjectures. For instance, we show that if P is a simplicial d-polytope that has no missing faces of dimension d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1), then the d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1)-skeleton of P and the set of sign vectors of affine i-stresses of P determine the combinatorial type of P. Along the way, we establish the partition of unity of affine stresses: for any d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1), the space of affine i-stresses of a simplicial d-polytope as well as the space of affine i-stresses of a simplicial d-12≤i≤d/2≥d-i+12≤i≤k≤d/2-1≥d-2i+2(d-1)2≤i≤d/2≥d-2i+2(i-1)1≤i≤(d-1)/2(d-1)-sphere (with a generic embedding) can be expressed as the sum of affine i-stress spaces of vertex stars. This is analogous to Adiprasito’s partition of unity of linear stresses for Cohen–Macaulay complexes.