maroon-scorch/computational-homology

A simple calculator for simplicial homology of finite Delta complexes using Smith normal forms. Final project of MATH 2410: Topology, instructed by Richard Schwartz.

Computational-Homology

For an explanation of the theory behind this calculator, please see main.pdf.

How to Run

To install dependencies (note this program was made in Python 3.7.3)

pip install -r requirements.txt

To compute the homology of 3-dimensional Lens Spaces:

python main.py lens <number of tetrahedrons>

To compute the homology on a custom Delta Complex:

python main.py <path to file>

Input Specification

We will explain how the input files should be formatted here, note that you can find some examples under the example folder.

The program takes a file of the following format:

• The first line of the file should be an non-negative integer indicating the number of vertices in the Delta Complex:

n

• The proceeding lines are as follows - The $i$-th line of file should contain a sequence of $i$-dimensional simplices, each separated by a semi-colon ;:

<simplex 1>; <simplex 2>; ...; <simplex k>

• For each simplex on line $i$, it should be represented by a list of indices, where the $j$-th index $v$ of the simplex represents the $j$-th face of the simplex. The index value $v$ means this face is the $v$-th simplex of line $i-1$:

<index_1> ... <index i>

For example, the following is the standard Delta Complex specification of the 2-torus:

1
0 0; 0 0; 0 0
0 1 2; 2 1 0

This structure only has 1 vertex, so all of its 3 edges start and end at the same point. Its two faces go in opposite orientations, hence their edges go in opposite directions.