On the cone of $f$-vectors of cubical polytopes

Abstract

What is the minimal closed cone containing all f -vectors of cubical d-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical g-vector coordinates, contains the nonnegative g-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold. R. M. Adin, D. Kalmanovich and E. Nevo On the cubical g-vectorDenote by e i the i -th unit vector in R ⌊d /2⌋ . Stacked cubical polytopes show that the ray spanned by e 1 is in C d , and neighborly cubical polytopes show that the ray spanned by e ⌊d /2⌋ is in C d . Our main result is that all the rays spanned by the vectors e i are in C d . ThusThe conjecture of Adin and Babson-Billera-Chan isAn analogue of Theorem 1 was previously known for the much wider class of PL cubical spheres [BBC97, Theorem 5.7]. Also, Conjecture 2 holds for d ≤ 5, by combining the constructions above with Steve Klee's result [Kle11, Prop.3.7] asserting that g c k (Q) ≥ 0 for any cubical polytope Q of dimension 2k + 1. Sanyal and Ziegler [SZ10] showed how to construct, from any simplicial (d − 2)-polytope P on n − 1 vertices and a total order v 1 < v 2 < . . . < v n−1 on its vertices, a cubical d-polytope Q = Q(P, <) on 2 n vertices; it is the projection of a deformed n-cube in R n onto the last d coordinates. Further, they showed that if P is k-neighborly then the k-skeleton of Q is isomorphic to the k-skeleton of the n-cube. We apply their construction to the case where P n is the k-neighborly k-stacked (d − 2)-polytope on n − 1 vertices constructed by McMullen and Walkup [MW71], with 1 ≤ k ≤ ⌊ d −2 2 ⌋, and with a suitable total order <. Analyzing the cubical g-vectors of the resulting polytopes Q(k, d, n) = Q(P n , <), as n tends to infinity, gives Theorem 1. See Theorem 13 and Corollary 14 for the exact values and asymptotic behavior of the cubical g-vectors.The generalized lower bound theorem for simplicial polytopes (GLBT), conjectured by McMullen and Walkup [MW71] and proved by Murai and Nevo [MN13], asserts that for 1 ≤ k < ⌊ d 2 ⌋, a simplicial dpolytope P is k-stacked if and only if g k+1 (P ) = 0. The polytopes Q(k, d, n) demonstrate that the natural cubical analogue of the GLBT does not hold:Theorem 3. For any k ≥ 1 and n ≥ d ≥ 2k + 4, we have g c k+2 (Q(k, d, n)) = 0, and Q(k, d, n) is not cubical (k + 1)-stacked.The paper is organized as follows. Basic definitions and notation are given in section 2. In section 3 we define our variant of the McMullen-Walkup polytopes. A sketch of the Sanyal-Ziegler construction is given in section 4. In section 5 we construct the polytopes Q(k, d, n) mentioned above and analyze their cubical g-vector. In section 6 we prove Theorem 3, showing that Q(k, d, n) is not (k + 1)-stacked. The final section 7 concludes with remarks and open questions.

Connectivity of cubical polytopes

“…6] exhibited cubical d-polytopes with the same ( d/2 − 1)-skeleton as the d -cube for every d > d, the so-called neighbourly cubical d-polytopes. And even more generally, Sanyal and Ziegler [13, p. 422], and later Adin, Kalmanovich and Nevo [1,Sec. 5], produced cubical d-polytopes with the same k-skeleton as the d -cube for every 1 ≤ k ≤ d/2 − 1 and every d > d, the so-called k-neighbourly cubical d-polytopes.…”

“…For d ≥ 4 provide precise values for the functions f (d) and g(d) or improve the lower and upper bounds in (1).…”

On the realization space of the cube

“…We give just a brief reminder of the definitions of a cubical d-polytope and its h c -vector and g c -vector. For more details, in particular, for the construction used in section 5, see [AKN19]. A d-polytope Q is cubical if each of its proper faces is combinatorially a cube.…”

“…In [AKN19], for each 1 ≤ i ≤ ⌊d/2⌋, the authors exhibit a sequences of cubical d-polytopes whose corresponding sequence of g c -vectors approaches the ray spanned by e i . This translates into sequences of f -vectors approaching the extremal rays of A d .…”

“…there exists a projective transformation φ and a d-tower T of at most 4d cubes, such that We apply this cubical connected sum operation to the cubical polytopes constructed recently in [AKN19]; the f -vectors of the latter approach the extremal rays of Adin's cone, which is conjectured to contain all f -vectors of d-polytopes [Adi96]. The following density result for f -vectors of cubical polytopes then follows: Let d denote the d-cube and f (P ) denote the f -vector of polytope P .…”