UW Combinatorics and Geometry Seminar: Hailun Zheng

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.