Preserving Hodge Vectors of Lattice Polytopes

TL;DR

This paper develops a method to compute and preserve the Hodge vectors of lattice polytopes, revealing new high-dimensional polytopes with identical Hodge vectors and introducing Lawrence twists as a key construction.

Contribution

It introduces a formula for the Hodge vector of Cayley polytopes, enabling the construction of infinitely many high-dimensional polytopes with the same Hodge vector, and presents Lawrence twists as a novel extension technique.

Findings

Computed Hodge vectors for Cayley polytopes.

Constructed infinitely many polytopes with identical Hodge vectors.

Provided new examples of thin polytopes and explained their properties.

Abstract

Given lattice polytopes $P_1, \ldots, P_k$ contained in a $k$-dimensional subspace $U \subseteq \mathbb{R}^d$ and a $d$-dimensional lattice polytope $Q \subset \mathbb{R}^d$, we compute the Hodge vector of the Cayley polytope $P_1 * \cdots * P_k * Q$, and show that it equals the mixed volume of $P_1, \ldots, P_k$ times the Hodge vector of the projection of $Q$ along $U$. Here, the Hodge vector of a lattice polytope is its local $h^*$-vector with leading and trailing zeroes removed. This result allows finding infinitely many high-dimensional lattice polytopes with the same Hodge vector that are not free joins. The proof relies on a closed formula for the Hodge-Deligne polynomial of generic complete intersections in the torus in terms of the bivariate/mixed $h^*$-polynomial. A special case of our construction is what we call Lawrence twists: extending the Gale transform by centrally-symmetric pairs of vectors. As applications, we can produce many new thin polytopes answering a question by Borger, Kretschmer and the second author, and we provide an alternative explanation of the thinness of $B_k$-polytopes answering a question of Selyanin.