Preserving Hodge vectors of lattice polytopes

In Ehrhart theory, the local h * -polynomial is a fundamental invariant of a lattice polytope. Its coefficient vector, the local h * -vector, is the coefficient vector of the top degree part of the Hodge-Deligne polynomial of the primitive cohomology with compact support of the associated generic hypersurface in the algebraic torus. In the literature on hypergeometric motives, the Hodge vector of a lattice polytope is its local h * -vector with leading and trailing zeroes removed. Of recent special interest are thin polytopes whose Hodge vectors vanish. Given lattice polytopes P 1 , ⋯ , P k contained in a k -dimensional subspace U ⊆ ℝ d and a d -dimensional lattice polytope P ⊂ ℝ d , we compute the Hodge vector of the Cayley polytope P 1 * ⋯ * P k * P , and show that it equals the mixed volume of P 1 , ... , P k times the Hodge vector of the projection of P along U . This 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.