Simplicial Complex Practice Problems

MATH  384
due Thursday, April 4

1. Write a function that returns the boundary of a simplex

The boundary of a k-dimensional simplex σ is the set of all (k-1)-dimensional faces of σ. Here, we are working with abstract simplices specified as lists of vertex indexes. For example, if σ={2,4,5}, then the boundary of σ is ∂σ={{2,4},{2,5},{4,5}}.

Write a function that computes the boundary of a given simplex. The input and output of your function should be as follows:
Input: A simplex σ, given as a list of vertex indexes.
Output: A list of all simplices in the boundary of σ.

2. Write a function that returns the coboundary of a simplex within a given complex

Suppose that simplicial complex K contains simplex σ. The coboundary of σ in K is the set of all simplices δ in K such that σ is in the boundary of δ. For example, let K={{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} and σ={1}. Then the coboundary of σ in K is {{1,2},{1,3}}.

Write a function that computes the coboundary of a given simplex in a simplicial complex. The input and output of your function should be as follows:
Input: A simplicial complex K (given as a list of simplices, each specified as a list of vertices), and a simplex σ
Output: A list of all simplices in the coboundary of σ in K.