A classical result by Pachner states that two d-dimensional combinatorial manifolds with boundary are PL homeomorphic if and only if they can be connected by a sequence of shellings and inverse ...shellings. We prove that for balanced, i.e., properly (d+1)-colored, manifolds such a sequence can be chosen such that balancedness is preserved in each step. As a key ingredient we establish that any two balanced PL homeomorphic combinatorial manifolds with the same boundary are connected by a sequence of basic cross-flips, as was shown recently by Izmestiev, Klee and Novik for balanced manifolds without boundary. Moreover, we enumerate combinatorially different basic cross-flips and show that roughly half of these suffice to relate any two PL homeomorphic manifolds.
Simplicial moves on balanced complexes Izmestiev, Ivan; Klee, Steven; Novik, Isabella
Advances in mathematics (New York. 1965),
11/2017, Volume:
320
Journal Article
Peer reviewed
Open access
We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly (d+1)-colored) triangulation of a combinatorial d-manifold into another balanced triangulation. These moves ...form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial d-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every m≥d+2 and any closed combinatorial d-manifold M, two m-colored triangulations of M can be connected by a sequence of bistellar flips that preserve the vertex colorings.
We prove a conjecture of Thomas Lam that the face posets of stratified spaces of planar resistor networks are shellable. These posets are called uncrossing partial orders. This shellability result ...combines with Lam's previous result that these same posets are Eulerian to imply that they are CW posets, namely that they are face posets of regular CW complexes. Certain subposets of uncrossing partial orders are shown to be isomorphic to type A Bruhat order intervals; our shelling is shown to coincide on these intervals with a Bruhat order shelling which was constructed by Matthew Dyer using a reflection order.
Our shelling for uncrossing posets also yields an explicit shelling for each interval in the face posets of the edge product spaces of phylogenetic trees, namely in the Tuffley posets, by virtue of each interval in a Tuffley poset being isomorphic to an interval in an uncrossing poset. This yields a more explicit proof of the result of Gill, Linusson, Moulton and Steel that the CW decomposition of Moulton and Steel for the edge product space of phylogenetic trees is a regular CW decomposition.
Subdivisions of shellable complexes Hlavacek, Max; Solus, Liam
Journal of combinatorial theory. Series A,
02/2022, Volume:
186
Journal Article
Peer reviewed
Open access
In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a ...property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the h-polynomial of a subdivision of a shellable complex. To apply it, we generalize the notion of reciprocal domains for convex embeddings of polytopes to abstract polytopes and use this generalization to define the family of stable shellings of a polytopal complex. We characterize the stable shellings of cubical and simplicial complexes, and apply this theory to answer a question of Brenti and Welker on barycentric subdivisions for the well-known cubical polytopes. We also give a positive solution to a problem of Mohammadi and Welker on edgewise subdivisions of cell complexes. We end by relating the family of stable line shellings to the combinatorics of hyperplane arrangements. We pose related questions, answers to which would resolve some long-standing problems while strengthening ties between the theory of interlacing polynomials and the combinatorics of hyperplane arrangements.
We prove the equivalence of EL-shellability and the existence of recursive atom ordering independent of roots. We show that a comodernistic lattice, as defined by Schweig and Woodroofe, admits a ...recursive atom ordering independent of roots, therefore is EL-shellable. We also present and discuss a simpler EL-shelling on one of the most important classes of comodernistic lattice, the order congruence lattices.
The edge-product space of phylogenetic trees is a regular CW complex whose maximal closed cells correspond to trivalent trees with leaves labeled by a finite set X. The face poset of this cell ...decomposition is isomorphic to the Tuffley poset, a poset of labeled forests, with a unique minimum adjoined. We show that the edge-product space of phylogenetic trees is gallery-connected. We then use combinatorial properties of the Tuffley poset and a related graph known as NNI-tree space to show that, although open intervals of the Tuffley poset were proven to be shellable by Gill, Linusson, Moulton, and Steel, the edge-product space is not shellable.
A generalization of Dowling lattices was recently introduced by Bibby and Gadish, in a work on orbit configuration spaces. The authors left open the question as to whether these posets are shellable. ...In this paper we prove EL-shellability and use it to determine the homotopy type. Our result generalizes shellability of Dowling lattices and of posets of layers of abelian arrangements defined by root systems. We also show that subposets corresponding to invariant subarrangements are not shellable in general.
This extended abstract is a summary of a recent paper which studies the enumeration of faces of subdivisions of cell complexes. Motivated by a conjecture of Brenti and Welker on the real-rootedness ...of the h-polynomial of the barycentric subdivision of the boundary complex of a convex polytope, we introduce a framework for proving real-rootedness of h-polynomials for subdivisions of polytopal complexes by relating interlacing polynomials to shellability via the existence of so-called stable shellings. We show that any shellable cubical, or simplicial, complex admitting a stable shelling has barycentric and edgewise subdivisions with real-rooted h-polynomials. Such shellings are shown to exist for well-studied families of cubical polytopes, giving a positive answer to the conjecture of Brenti and Welker in these cases. The framework of stable shellings is also applied to answer to a conjecture of Mohammadi and Welker on edgewise subdivisions in the case of shellable simplicial complexes.
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