This talk will review lots of background in the area of topological combinatorics along the way towards discussing a new technique for studying the topological structure of simplicial complexes known as order complexes of partially ordered sets. Our main results give a new twist on a popular tool of poset topology called lexicographic shellability. When Bjoerner and Wachs introduced one of the main forms of lexicographic shellability, namely CL-shellability, they also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering. We generalize the notion of recursive atom ordering, and we prove that any such generalized recursive atom ordering may be transformed via a reordering process into a recursive atom ordering. We also prove that a finite bounded poset admits a generalized recursive atom ordering if and only if it is ``CC-shellable'' in a way that is self-consistent in a certain sense. This allows us to conclude that CL-shellability is equivalent to self-consistent CC-shellability. As an application, we prove that the uncrossing orders, namely the face posets for stratified spaces of planar electrical networks, are dual CL-shellable. This is joint work with Grace Stadnyk.

(This talk is rescheduled from 4/14 due to logistical issues.)