Research

Abstract: Bier spheres arise as deleted joins of simplicial complexes with their combinatorial Alexander duals and form one of the largest known families of simplicial spheres. We study centrally symmetric Bier spheres and give a simple criterion for when they cannot arise as boundaries of centrally symmetric polytopes. From this, we obtain a large new family of simplicial polytopes with combinatorial automorphisms that cannot be realized geometrically. Prior to our construction, the Bokowski--Ewald--Kleinschmidt polytope was the only known simplicial example exhibiting these properties. By Smith theory, these polytopes have noncontractible realization spaces. Finally, we establish that every Bier sphere with at most 12 vertices is polytopal.

Abstract: Hypergraphic polytopes ∆_H arise as Minkowski sums of simplices indexed by the hyperedges of a hypergraph H. Orienting the 1-skeleton of such a polytope by a certain generic linear functional gives rise to the hypergraphic poset P_H. Hypergraphic posets include the weak order for the permutahedron and the Tamari lattice for the associahedron. This motivates the problem of determining when P_H is a lattice. In this paper, we give a complete lattice characterization for cyclic interval hypergraphs, extending the result of Bergeron and Pilaud for interval hypergraphs, and the result of Adenbaum et al. for the complete cyclic interval hypergraph.

Abstract: We present a 6-dimensional centrally symmetric simplicial polytope for which the antipodal quotient of its boundary forms a 24-vertex triangulation of the 5-dimensional real projective space. This 6-polytope is highly symmetric with an automorphism group of order 192, and is of independent interest. We conjecture that our construction uses the fewest number of vertices among all triangulations of RP^5. Our method also produces two triangulations of RP^6 on 45 and 49 vertices; both improve the previously best known construction in dimension 6 that used 53 vertices.

Abstract: In 2022 Kim showed when a graph G is ternary (without induced cycles of length divisible by three), its independence complex Ind(G) is either contractible or homotopy equivalent to a sphere. In this paper, we show that when Ind(G) is homotopy equivalent to a sphere of dimension dim Ind(G), the complex is Gorenstein. Equivalently, G is a 1-well-covered graph. This answers a question by Faridi and Holleben.

We then focus on the independence complexes of Gorenstein planar ternary graphs. We prove that they are boundaries of vertex decomposable simplicial polytopes. We show that the transformations among these flag spheres using edge subdivisions and contractions can be modeled by the Hasse diagram of the partition refinement poset. In addition, their h-polynomials are products of Delannoy polynomials and thus real-rooted. Finally, we demonstrate a way to construct nonplanar Gorenstein (1-well-covered) ternary graphs from planar ones.

Abstract: We show that the facet-ridge graph of a shellable simplicial sphere Δ uniquely determines the entire combinatorial structure of Δ. This generalizes the celebrated result due to Blind and Mani (1987), and Kalai (1988) on reconstructing simple polytopes from their graphs. Our proof utilizes the notions of good acyclic orientations from Kalai's proof as well as k-systems introduced by Joswig, Kaibel, and Körner.

Abstract: Nevo, Santos, and Wilson constructed 2^Ω(N^d) combinatorially distinct simplicial (2d−1)-spheres with N vertices. We prove that all spheres produced by one of their methods are shellable. Combining this with prior results of Kalai, Lee, and Benedetti and Ziegler, we conclude that for all D>=3, there are 2^Θ(N^⌈D/2⌉) shellable simplicial D-spheres with N vertices.