```
chomp::Homology H;
chomp::Cube *c = new chomp::Cube ("[0,0]");
H . add (c);
```

How would you install CHomP on a Linux system?

You would first download the source code from the CHomP website. Then, you would extract the files, navigate to the directory in the terminal, and run the 'make' command to compile the code. Finally, you would run 'make install' to install the program.

What are the main components of CHomP?

The main components of CHomP are its libraries for computing homology and its command-line tools for processing input data and displaying output.

Describe the difference between homology and cohomology in CHomP.

Homology is the study of spaces by associating a group to each dimension, which captures the 'holes' in that dimension. Cohomology is a dual concept to homology, and it captures the 'holes' in the complementary dimensions.

How would you compute the homology of a simplicial complex using CHomP?

You would use the 'homcubes' command-line tool, providing it with a file that describes the simplicial complex. The tool would then output the Betti numbers, which describe the homology of the complex.

What are Betti numbers in the context of CHomP?

Betti numbers are integers that describe the homology of a space. The nth Betti number is the rank of the nth homology group, which counts the number of 'n-dimensional holes' in the space.

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Describe the difference between absolute and relative homology in CHomP.

Absolute homology is the homology of a space on its own, while relative homology is the homology of a space relative to a subspace. In CHomP, you can compute relative homology by providing two input files to the 'homcubes' tool.

How would you compute the cohomology of a cubical complex using CHomP?

You would use the 'chmcubes' command-line tool, providing it with a file that describes the cubical complex. The tool would then output the Betti numbers, which describe the cohomology of the complex.

What are cubical complexes in the context of CHomP?

Cubical complexes are a type of topological space that can be described using cubes of various dimensions. They are an alternative to simplicial complexes, and can be more convenient for certain types of data.

Describe the difference between the 'homcubes' and 'chmcubes' tools in CHomP.

'homcubes' is used to compute the homology of a simplicial complex, while 'chmcubes' is used to compute the cohomology of a cubical complex. They are similar in function but operate on different types of input data.

How would you use CHomP to compute the persistent homology of a filtration?

You would use the 'homcubes' tool with the '-p' option, providing it with a file that describes the filtration. The tool would then output the persistence diagram, which describes the persistent homology of the filtration.

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What does this simple piece of CHomP code do?

```
chomp::Homology H;
chomp::Cube *c = new chomp::Cube ("[0,0]");
H . add (c);
```

This code creates a new instance of the Homology class, creates a new Cube with the coordinates [0,0], and then adds this cube to the Homology instance.

What will be the output of this CHomP code?

```
chomp::Cube c ("[0,0]");
std::cout << c . dimension ();
```

This code will output '2'. The dimension method of the Cube class returns the dimension of the cube, which in this case is 2.

What does this CHomP code do with the array of Cubes?

```
chomp::Cube cubes[3] = {chomp::Cube ("[0,0]"), chomp::Cube ("[1,1]"), chomp::Cube ("[2,2]")};
chomp::Homology H;
for (int i = 0; i < 3; i++) {
H . add (&cubes[i]);
}
```

This code creates an array of three Cubes with different coordinates, creates a new Homology instance, and then adds each Cube in the array to the Homology instance.

What does this CHomP code do related to threading?

```
chomp::Homology H;
std::thread t1([&](){ H . add (new chomp::Cube ("[0,0]")); });
std::thread t2([&](){ H . add (new chomp::Cube ("[1,1]")); });
t1 . join ();
t2 . join ();
```

This code creates a new Homology instance and two threads. Each thread adds a new Cube with different coordinates to the Homology instance. The join method is then called on each thread, which causes the main thread to wait for the two threads to finish execution.

What is persistent homology in the context of CHomP?

Persistent homology is a method for computing the homology of a filtration, which is a nested sequence of spaces. It captures the 'lifespan' of homology classes as the filtration progresses.

Describe the difference between the 'homcubes' and 'homchain' tools in CHomP.

'homcubes' is used to compute the homology of a simplicial or cubical complex, while 'homchain' is used to compute the homology of a chain complex. They are similar in function but operate on different types of input data.

How would you use CHomP to compute the homology of a chain complex?

You would use the 'homchain' tool, providing it with a file that describes the chain complex. The tool would then output the Betti numbers, which describe the homology of the complex.

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