Buchstaber Invariant of Simple Polytopes

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The problem stated by Victor M. Buchstaber is to find a simple combinatorial description of an s-number. We describe the main properties of s(P) and study the properties of simple n-polytopes with n+3 facets. In particular, we find the value of an s-number for such polytopes, a simple formula for their h-polynomials and the bigraded cohomology rings of the corresponding moment-angle complexesComment: 25 pages; typos corrected; references adde

Geometric and Combinatorial Properties of Lattice Polytopes Defined from Graphs

Polytopes are geometric objects that generalize polygons in the plane and polyhedra in 3-dimensional space. Of particular interest in geometric combinatorics are families of lattice polytopes defined from combinatorial objects, such as graphs. In particular, this dissertation studies symmetric edge polytopes (SEPs), defined from simple undirected graphs. In 2019, Higashitani, Jochemko, and Michalek gave a combinatorial description of the hyperplanes that support facets of a symmetric edge polytope in terms of certain labelings of the underlying graph.Using this framework, we explore the number of facets that can be attained by the symmetric edge polytopes for graphs with certain structure. First, we establishformulas or bounds for the number of facets attained by families of sparse, connected graphs, and give conjectures concerning the maximum and minimum facet counts for more general families. We also consider the number of facets of SEPs arising from graphs generated by several random graph models and investigate a conjectured connection between facet counts for SEPs and clustering metrics on their underlying graphs

Semiregular Polytopes and Amalgamated C-groups

In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper studies semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices. We analyze the structure of the automorphism group, focusing in particular on polytopes with two kinds of regular facets occurring in an "alternating" fashion. In particular we use group amalgamations to prove that given two compatible n-polytopes P and Q, there exists a universal abstract semiregular (n+1)-polytope which is obtained by "freely" assembling alternate copies of P and Q. We also employ modular reduction techniques to construct finite semiregular polytopes from reflection groups over finite fields.Comment: Advances in Mathematics (to appear, 28 pages

On Compact Hyperbolic Coxeter Polytopes with Few Facets

This thesis is concerned with classifying and bounding the dimension of compact hyperbolic Coxeter polytopes with few facets. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. We use this to complete the classification of d-polytopes with d+4 facets for d=4 and 5. In dimensions 4 and 5, there are 341 and 50 polytopes, respectively, yielding many new examples for further study. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is d=6. We furthermore show that any polytope of dimension 6 must have a missing face of size 3 or 4. The second portion of this thesis provides new upper bounds on the dimension of compact hyperbolic Coxeter d-polytopes with d+k facets for k = 5. In the process of proving the present bounds, we additionally show that there are no compact hyperbolic Coxeter 3-free polytopes of dimension higher than 13. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets

Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes

This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindst\"om. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses the following result by Below. Given any polytope with vertices having algebraic coordinates, there is a combinatorial "stamp" polytope with a specified face that is projectively equivalent to the given polytope in all realizations. Here we construct a new stamp polytope that is closely related to Richter-Gebert's proof of universality for 4-dimensional polytopes, and we generalize several tools from that proof