## The antiprism triangulation

## Martina Juhnke-Kubitzke

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 iterated crossings of the faces of \(\Delta\), from a geometric point of view.

This talk studies enumerative invariants and algebraic properties associated with this triangulation, such as the transformation of the \(h\)-vector of \(\Delta\) under antiprism triangulation. The main results show 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 \(\Delta\). This is joint work with Christos Athanasiadis and Jan-Marten Brunink.