Filtrations and Signatures Simplicial Complexes

K −∞ is the empty set, K0 = K, and K∞ = D is the dual of the Voronoï diagram, also known as the Delaunay triangulation of S Delaunay, 1934. It is easy to see that the Voronoï regions do not change and simplices are only added as the balls are expanded. Therefore, K α 1 ⊆ Kα 2 for α 1 ≤ α 2 . This implies that the α-complex provides a filtration of the Delaunay triangulation of S. This filtration gives a partial ordering on the simplices of K. For each simplex σ ∈ D, there is a unique birth time α 2 σ such that σ ∈ Kα iff α 2 ≥ α 2 σ. We order the simplices such that α 2 σ α 2 τ implies σ precedes τ in the ordering. More than one simplex may be born at a time, and such cases may arise even if S is in general position. For example, in Figure 2.16, edge uw is born at the same moment as triangle uvw. As noted before, we may convert this partial ordering into a total ordering easily. In fact, for α-shape filtrations, we always do so, allowing only a single simplex to enter the complex at any time. In Figure 2.18, we show a few complexes in an alpha-complex filtration for a small protein, Gramicidin A. We have seen this protein before, first modeled as a molecular surface in Figure 1.5a, and then as a van der Waals surface in Figure 2.15b. Note that the alpha-complex model has many additional topological attributes at different times in the filtration. One of the main results of this book is the identification of the significant topological features from these attributes.

2.5 Manifold Sweeps

Alpha-shapes allow us to explore the shape of finite point sets and unions of balls. In addition to such spaces, we are interested in exploring manifolds with height functions. In Example 1.10, we saw how the geometry of a manifold dictates the topology of its iso-lines. We use this example to motivate another geometrically ordered filtration in this section, postponing theoretical justifi- cation for it until we have been introduced to Morse Theory in Chapter 5. Let K be a triangulation of a compact 2-manifold without boundary M. Let h : M → R be a function that is linear on every triangle. The function is de- fined, consequently, by its values at the vertices of K. We will assume that h u = hv for all vertices u = v ∈ K. Again, simulation of simplicity is the computational justification for this assumption Edelsbrunner and Mücke, 1990. It is common to refer to h as the height function, because it matches our intuition of a geographic landscape. One needs to be careful, however, not to allow the intuition to limit one’s imagination, as h can be any continuous function. In a simplicial complex, the natural concept of a neighborhood of a vertex