Homology Group Calculator
Homology Group Calculator computes the homology groups \\( H_0 \\), \\( H_1 \\), and \\( H_2 \\) of a simplicial complex defined by vertices, edges, and triangles. Enter vertices (e.g., A, B, C), edges (e.g., AB, BC, CA), and optionally triangles (e.g., ABC) as comma-separated lists. The calculator computes the groups using algebraic topology methods, shows steps with MathJax, and visualizes the complex with p5.js.
Homology Group Calculator
This calculator computes the homology groups \\( H_0 \\), \\( H_1 \\), and \\( H_2 \\) of a simplicial complex defined by vertices, edges, and optionally triangles. Input vertices (e.g., A, B, C), edges (e.g., AB, BC, CA), and triangles (e.g., ABC) as comma-separated lists. The tool calculates \\( H_0 \\) (connected components), \\( H_1 \\) (loops), and \\( H_2 \\) (voids) using algebraic topology methods. Steps are shown with MathJax, and the complex is visualized with p5.js. Note: Limited to complexes with up to 10 vertices and dimension 2 for computational feasibility.
Example 1: Single Vertex
Input: Vertices = A, Edges = (empty), Triangles = (empty).
Result: \\( H_0 = \mathbb{Z}, H_1 = 0, H_2 = 0 \\).
Meaning: A single vertex forms one connected component.
Example 2: Triangle
Input: Vertices = A, B, C, Edges = AB, BC, CA, Triangles = ABC.
Result: \\( H_0 = \mathbb{Z}, H_1 = 0, H_2 = \mathbb{Z} \\).
Meaning: One connected component, no loops (filled triangle), one 2-dimensional void.
Example 3: Disconnected Graph
Input: Vertices = A, B, C, D, Edges = AB, CD, Triangles = (empty).
Result: \\( H_0 = \mathbb{Z}^2, H_1 = 0, H_2 = 0 \\).
Meaning: Two connected components, no loops, no voids.