Combinatorial persistent homology transform

Figure 1.  The V, embedded in $ \mathbb{R}^2 $. This simplicial complex has three vertices and two edges. By exploring the combinatorial PH transform for this example, we illustrate each step of the construction

Figure 2.  In (a), we see the three linear subpaces of $ \mathbb{R}^2 $ that are used to define the cellulation over $ \mathbb{S}^1 $. In (b), each cell is labeled by a vector in $ \{-, 0, +\}^3 $ according to which side of $ S_{1, 2} $, $ S_{1, 3} $, and $ S_{2, 3} $ the cell falls. For example, the vector $ (-++) $ labels the one-cell whose points are all in $ S_{1, 2}^- $, $ S_{1, 3}^+ $, and $ S_{2, 3}^+ $. The vector $ (0++) $ labels the zero-cell that is in $ S_{1, 2} $, $ S_{1, 3}^+ $, and $ S_{2, 3}^+ $. In fact, $ \mathit{C}_{(0++)} = S_{1, 2}\cap S_{1, 3}^+\cap S_{2, 3}^+ $. Note that no label is $ (000) $, and that all labels are distinct. The partial order of the cells is denoted by arrows (where $ a \to b $ indicates that $ b < a $)

Figure 3.  The bounded lattice functions for the three highlighted face relations in Figure 2(b). Notice that the two maps into $ \mathit{P}_{(0++)} $ are nearly bijections, except for two vertices ($ v_1 $ and $ v_2 $) that map to the same equivalence class for both maps. This corresponds to the transposition of the two vertices between directions in $ \mathit{C}_{(+++)} $ and $ \mathit{C}_{(-++)} $. In fact, this property holds more generally for any face relation between codimension-one cells

Figure 4.  The maps $ \overline{P_{(+++)}} \to \overline{P_{0++}} $ and $ \mathtt{ZB}_0( \overline{P_{(+++)}} \to \overline{P_{0++}}) $. In both maps, the objects in the same pink region get mapped to the same object in the codomain. Using the field $ \mathtt{k} = \mathbb{Z} / 2 \mathbb{Z} $, for $ b\neq \top $, $ \mathtt{ZB}_0[a, b] $ counts the number of zero-cycles in the simplicial complex $ \mathit{F}(a) $ that are zero-boundaries in the larger simplicial complex $ \mathit{F}(b) $. $ \mathtt{ZB}_0[a, \top] $ is a count of the number of vertices in $ \mathit{F}(a) $

Figure 5.  A geometric complex in $ \mathbb{R}^3 $ with the induced cellulation of $ \mathbb{S}^2 $. The cellulation depends only on the vetices of the complex. Since the vertices are in general position, each great circle on $ \mathbb{S}^2 $ is distinct

Figure 6.  Face relations induce surjective poset maps. These maps, in turn, induce arrows in $ \mathtt{Fil} $