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