The Depth Poset under Transpositions in the Filter

TL;DR

This work studies how transpositions in a filter affect the depth poset of a filtered Lefschetz complex. It provides a complete case analysis—BB, DD, and BD switches—together with efficient matrix-reduction methods to compute the related birth/death relations and their transitive closures. Computational experiments on random PL functions and straight-line homotopies quantify depth-poset size, component structure, and sensitivity, showing BD-type switches are rare and that the transitive reduction is relatively stable. The results motivate extending depth posets to discrete vector fields and stochastic models and linking to topology-optimization algorithms.

Abstract

The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in the present work and for updating the persistence diagram under transpositions (Vineyard persistence), we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.

Figures (12)

• Figure 1: Upper left: a complex with four vertices, five edges, three triangles ($\alpha, \beta, \gamma$, in which $\alpha$ and $\beta$ have the same boundary), and one $3$-cell ($\Sigma$, which is sandwiched between $\alpha$ and $\beta$). Lower left: the ordering implied by a filter, with birth-death pairs as indicated (see definition below). Right: the death and birth relations on the birth-death pairs as computed by Algorithms 1 and 2 below.
• Figure 2: The three types of switches. From left to right: swapping two birth-giving cells (BB-type), swapping two death-giving cells (DD-type), and swapping a birth-giving with a death-giving cell (BD-type). In the first two cases, the intervals that correspond to the pairs are nested---before and after the switch---and in the third case, they are disjoint---again before and after the switch.
• Figure 3: To cancel $a \prec b$, we connect all facets of $b$ to all cofacets of $a$. The remainder of the face relation is unchanged.
• Figure 4: The only difference to the example in Figure \ref{['fig:Example']} is the order of the edges $\tt c$ and $\tt d$. To the right, we see the death and birth relations, which are both empty.
• Figure 5: The configurations before and after the transposition of two birth-giving cells. From left to right: Cases I.1, I.2, I.3, with the configuration before and after the transposition at the top and the bottom, respectively, or vice versa for the inverse transposition. Within each panel, we show the relevant birth- and death-giving cells as rows and columns of the boundary matrix, respectively, the two birth-death pairs as bars with dotted face relation, and the corresponding nodes and arcs in the death and birth relations.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Lemma 2.1
• Definition 3
• Definition 4
• Theorem 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3