Shellability of face posets of electrical networks and the CW poset property

TL;DR

This paper proves that face posets of planar resistor network stratifications are shellable and CW posets, connecting combinatorial properties with topological structures, and extends results to phylogenetic tree spaces.

Contribution

It establishes shellability and CW poset properties for face posets of electrical network stratifications, linking them to Bruhat order intervals and phylogenetic tree spaces.

Findings

Face posets of planar resistor networks are shellable.

These posets are CW posets, i.e., face posets of regular CW complexes.

Shellings coincide with Bruhat order shellings on certain intervals.

Abstract

We prove a conjecture of Thomas Lam that the face posets of stratified spaces of planar resistor networks are shellable. These posets are called uncrossing partial orders. This shellability result combines with Lam's previous result that these same posets are Eulerian to imply that they are CW posets, namely that they are face posets of regular CW complexes. Certain subsets of uncrossing partial orders are shown to be isomorphic to type A Bruhat order intervals; our shelling is shown to coincide on these intervals with a Bruhat order shelling which was constructed by Matthew Dyer using a reflection order. Our shelling for uncrossing posets also yields an explicit shelling for each interval in the face posets of the edge product spaces of phylogenetic trees, namely in the Tuffley posets, by virtue of each interval in a Tuffley poset being isomorphic to an interval in an uncrossing poset. This yields a more explicit proof of the result of Gill, Linusson, Moulton and Steel that the CW decomposition of Moulton and Steel for the edge product space of phylogenetic trees is a regular CW decomposition.