Topological invariants from simplicial complexes
β₀ = connected components, β₁ = independent loops/cycles, β₂ = enclosed voids. Computed via boundary matrices and rank-nullity. The Euler characteristic χ = β₀ − β₁ + β₂. Vietoris-Rips: connect vertices within radius r to see topological features appear and disappear as r grows (persistence!).