Rank Selection and Depth Conditions for Balanced Simplicial Complexes

We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that if a balanced simplicial complex satisfies Serre's condition $(S_{\ell})$ then so do all of its rank selected subcomplexes. We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial complex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes is expensive because of a combinatorial explosion in the complex size. For n points in $$\mathbb {R}^d$$ R d , we present a scheme to construct a 2-approximation of the filtration of the Rips complex in the $$L_\infty $$ L ∞ -norm, which extends to a $$2d^{0.25}$$ 2 d 0.25 -approximation in the Euclidean case. The k-skeleton of the resulting approximation has a total size of $$n2^{O(d\log k +d)}$$ n 2 O ( d log k + d ) . The scheme is based on the integer lattice and simplicial complexes based on the barycentric subdivision of the d-cube. We extend our result to use cubical complexes in place of simplicial complexes by introducing cubical maps between complexes. We get the same approximation guarantee as the simplicial case, while reducing the total size of the approximation to only $$n2^{O(d)}$$ n 2 O ( d ) (cubical) cells. There are two novel techniques that we use in this paper. The first is the use of acyclic carriers for proving our approximation result. In our application, these are maps which relate the Rips complex and the approximation in a relatively simple manner and greatly reduce the complexity of showing the approximation guarantee. The second technique is what we refer to as scale balancing, which is a simple trick to improve the approximation ratio under certain conditions.

Deformation Cones of Nested Braid Fans

Generalized permutohedra are deformations of regular permutohedra and arise in many different fields of mathematics. One important characterization of generalized permutohedra is the Submodularity Theorem, which is related to the deformation cone of the Braid fan. We lay out general techniques for determining deformation cones of a fixed polytope and apply it to the Braid fan to obtain a natural combinatorial proof for the Submodularity Theorem. We also consider a refinement of the Braid fan, called the nested Braid fan, and construct usual (respectively, generalized) nested permutohedra that have the nested Braid fan as (respectively, a coarsening of) their normal fan. We extend many results on generalized permutohedra to this new family of polytopes, including a one-to-one correspondence between faces of nested permutohedra and chains in ordered partition posets, and a theorem analogous to the Submodularity Theorem. Finally, we show that the nested Braid fan is the barycentric subdivision of the Braid fan, which gives another way to construct this new combinatorial object.

Fast Computation of Polynomial Data Points Over Simplicial Face Values

Polynomial functions [Formula: see text] of degree [Formula: see text] have a form in the Bernstein basis defined over [Formula: see text]-dimensional simplex [Formula: see text]. The Bernstein coefficients exhibit a number of special properties. The function [Formula: see text] can be optimised by the smallest and largest Bernstein coefficients (enclosure bounds) over [Formula: see text]. By a proper choice of barycentric subdivision steps of [Formula: see text], we prove the inclusion property of Bernstein enclosure bounds. To this end, we provide an algorithm that computes the Bernstein coefficients over subsimplices. These coefficients are collected in an [Formula: see text]-dimensional array in the field of computer-aided geometric design. Such a construct is typically classified as a patch. We show that the Bernstein coefficients of [Formula: see text] over the faces of a simplex coincide with the coefficients contained in the patch.

An integer sequence and standard monomials

For an (oriented) graph [Formula: see text] on the vertex set [Formula: see text] (rooted at [Formula: see text]), Postnikov and Shapiro (Trans. Amer. Math. Soc. 356 (2004) 3109–3142) associated a monomial ideal [Formula: see text] in the polynomial ring [Formula: see text] over a field [Formula: see text] such that the number of standard monomials of [Formula: see text] equals the number of (oriented) spanning trees of [Formula: see text] and hence, [Formula: see text], where [Formula: see text] is the truncated Laplace matrix of [Formula: see text]. The standard monomials of [Formula: see text] correspond bijectively to the [Formula: see text]-parking functions. In this paper, we study a monomial ideal [Formula: see text] in [Formula: see text] having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal [Formula: see text] is the cellular resolution supported on a subcomplex of the first barycentric subdivision [Formula: see text] of an [Formula: see text] simplex [Formula: see text]. The integer sequence [Formula: see text] has many interesting properties. In particular, we obtain a formula, [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], similar to [Formula: see text].

4-Colored Triangulation of 3-Maps

We describe an algorithm to triangulate a general 3-dimensional-map on an arbitrary space in such way that the resulting 3-dimensional triangulation is vertex-colorable with four colors. (Four-colorable triangulations can be efficiently represented and manipulated by the GEM data structure of Montagner and Stolfi.) The standard solution to this problem is the barycentric subdivision (BCS) of the map. Our algorithm yields a 4-colored triangulation that is provably smaller than the BCS, and in practice is often a small fraction of its size. When the input map is a shellable triangulation of a 3-ball, in particular, we can prove that the output size is less than [Formula: see text] times the size of the BCS.