Combinatorics of Antiprism Triangulations

Combinatorics of Triangulations and Hilbert Series

We introduce the basic concepts of Gröbner basis theory and its relations to polytope theory. This will cover very basic parts of [1], the basic chapters from [3] and the first chapters from [11]. In particular, we will define the Gröbner fan, which will play a major role in some of the research problems we will pose later. In this lecture we relate triangulations to Gröbner bases and Hilbert-s...

Combinatorics of Triangulations Title: Associate Professor of Information Science

Several combinatorial aspects of triangulations and their generalizations are studied in this thesis. A triangulation of a point con guration and a d-dimensional polyhedron whose vertices are among the points is a decomposition of the polyhedron using d-simplices with vertices among the points. The two main elds triangulations appear are combinatorial geometry in mathematics and computational g...

Two problems in geometric combinatorics: Efficient triangulations of the hypercube

Grand Antiprism and Quaternions

Vertices of the 4-dimensional semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E8 root system which decomposes into two copies of the root system of H4. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter groupW (H4). It is the groupAut(H2⊕...

Combinatorics of Triangulations and the Chern-Simons Invariant for Hyperbolic 3-Manifolds

In this paper we prove some results on combinatorics of triangulations of 3-dimensional pseudo-manifolds, improving on results of [NZ], and apply them to obtain a simplicial formula for the Chern-Simons invariant of an ideally triangulated hyperbolic 3-manifold. Combining this with [MN] gives a simplicial formula for the η invariant also. In effect, the main ingredient in the formula is the sum...