Note on combinatorial structure of self-dual simplicial complexes

Abstract

Simplicial complexes $K$, in relation to their Alexander dual $\widehat{K}$ , can be classified as self-dual ($K=\widehat{K}$), sub-dual ($K\subseteq \widehat{K}$), super-dual ($K\supseteq \widehat{K}$), or transcendent (neither sub-dual nor super-dual). We explore a connection between sub-dual and self-dual complexes providing a new insight into combinatorial structure of self-dual complexes. The {\em root operator} associates with each self-dual complex $K$ a sub-dual complex $\surd K$ on a smaller number of vertices. We study the operation of {\em minimal restructuring} of self-dual complexes and the properties of the associated {\em neighborhood graph}, defined on the set of all self-dual complexes. Some of the operations and relations, introduced in the paper, were originally developed as a tool for computer-based experiments and enumeration of self-dual complexes.