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Computing…
Scale: r is the ball radius. In the Rips complex an edge enters at r = ½·distance; in the alpha complex each simplex enters at its alpha radius. Hover a bar or a dot to read its values. Click a bar to jump to the radius where you clicked (a dot jumps to the middle of its life) and highlight a representative in the point cloud: for H₁, the shortest loop at that radius that its cocycle detects; for H₀, the points of that cluster. Click empty space in a chart to clear the highlight.
H₀ with k > 1 uses cluster persistence (Bobrowski & Skraba, Entropy 2023): a component counts as a cluster only once it holds at least k points, so tiny components never get a bar, and a cluster's bar starts when it reaches size k. Grey points in the cloud belong to components still smaller than k.