Betti Numbers
Topological invariants of simplicial complexes: β₀ = components, β₁ = loops, β₂ = voids
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Point
Segment
Triangle
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Circle
Torus*
Sphere*
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Betti Numbers
0
β₀
components
0
β₁
loops
0
β₂
voids
χ = β₀ − β₁ + β₂ = 0
Vertices:
0
| Edges:
0
| Faces:
0
Boundary Matrix ∂₁ (columns=edges, rows=vertices)
No edges yet
Formula:
χ = V − E + F
β₀ = rank H₀ = components
β₁ = E − V + β₀ − rank(∂₂)
β₂ = rank H₂ = enclosed voids