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arXiv · 2006.10789

Combinatorics of antiprism triangulations

Christos A. Athanasiadis·Jan-Marten Brunink·Martina Juhnke-Kubitzke
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Abstract

The antiprism triangulation provides a natural way to subdivide a simplicial complex $\Delta$, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as the simplicial complex of chains of multi-pointed faces of $\Delta$, from a combinatorial point of view, and by successively applying the antiprism construction, or balanced stellar subdivisions, on the faces of $\Delta$, from a geometric point of view. This paper studies enumerative invariants associated to this triangulation, such as the transformation of the $h$-vector of $\Delta$ under antiprism triangulation, and algebraic properties of its Stanley--Reisner ring. Among other results, it is shown that the $h$-polynomial of the antiprism triangulation of a simplex is real-rooted and that the antiprism triangulation of $\Delta$ has the almost strong Lefschetz property over ${\mathbb R}$ for every shellable complex $\Delta$. Several related open problems are discussed.

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Christos A. Athanasiadis, Jan-Marten Brunink, Martina Juhnke-Kubitzke. 2020-06-18. Combinatorics of antiprism triangulations. https://arxiv.org/abs/2006.10789

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