euler_characteristic: Compute the Euler characteristic \(\chi\) of a simplicial complex
Description
Compute the Euler characteristic \(\chi\) of a simplicial complex
Usage
euler_characteristic(simplices, tol)
Value
An integer representing the Euler characteristic \(\chi\).
Arguments
simplices
A list of simplices (each a numeric vector).
tol
Optional numerical tolerance to pass to rankMatrix().
Details
The Euler characteristic is computed as:
$$\chi = \sum_{k=0}^{k_{\max}} (-1)^k \beta_k$$
where \(\beta_k\) is the \(k\)th Betti number, and \(k_{\max}\) is the highest dimension of any simplex in the complex.
Interpretation of values:
\(\chi = 2\): Sphere-like surfaces
\(\chi = 1\): Disk-like spaces
\(\chi = 0\): Torus-like or circle-like spaces
\(\chi < 0\): Surfaces with multiple handles or genus