The merging operation and (d-i)-simplicial i-simple d-polytopes

Abstract:

A $d$-polytope is called $(d-i)$-simplicial if all of its $(d-i)$-faces are simplices. It is $i$-simple if every $(d-i-1)$-face belongs to exactly $i+1$ facets. A few low-dimensional examples of $(d-i)$-simplicial $i$-simple polytopes arising from regular polytopes are known. For $d>3$, a $d$-dimensional demicube and its dual are $3$-simplicial $(d-3)$-simple and $(d-3)$-simplicial $3$-simple, respectively. In addition to these finitely many examples, Paffenholz and Ziegler proved the existence of infinite families of $(d-2)$-simplicial $2$-simple $d$-polytopes for all $d>3$. However, for general $i>4$ and $d>2i-1$, it is not known whether non-simplex $(d-i)$-simplicial $i$-simple $d$-polytopes exist.

Given two $d$-polytopes $P$ and $Q$, where $P$ has a simplex facet $F$ and $Q$ has a simple vertex, we define an operation called the merge of $P$ and $Q$ along $F$ and $v$. We show that if for some $0<i<d$, both $P$ and $Q$ are $(d-i)$-simplicial $i$-simple, then the merge of $P$ and $Q$ is also $(d-i)$-simplicial $i$-simple. We then use this operation to construct infinite families of $i$-simplicial $i$-simple $2i$-polytopes for all $i<5$. Furthermore, infinitely many of these polytopes have another nice property: they are self-dual.

Joint work with Isabella Novik.

Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.

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Meeting ID: 915 4733 5974