April 4 Practice Problems

MATH  384
due Tuesday, April 9

1. Write a function that returns the Euler characteristic of a simplicial complex

The Euler characteristic of a simplicial complex K is integer χ(K)=(∑_(i=0)^dimK)(((-1))^i)(Subscript[K, i]), where (Subscript[K, i]) is the number of i-dimensional simplices in K. For example, if K={{1},{2},{3},{4},{1,2},{2,3},{2,4},{3,4},{2,3,4}}, then χ(K)=4-4+1=1 since K contains four 0-dimensional simplices, four 1-dimensional simplices, and one 2-dimensional simplex.

Write a function that computes the Euler characteristic of a given simplicial complex. The input and output of your function should be as follows:
Input: A abstract simplicial complex K.
Output: The Euler characteristic χ(K).

2. A number-theoretic simplicial complex

Let S be the set of all square-free integers n>=2, meaning integers greater than 1 that are not divisible by any square. That is, S={2,3,5,6,7,10,11,13,14,15,17,...}. For each integer n>=1, let K(n) be the simplicial complex defined on the vertex set V(n)={s∈S | s <=n} by the following rule. For each k>0, the k-simplices are subsets {(Subscript[s, 0]),(Subscript[s, 1]),...,(Subscript[s, k])}⊂V(n) of vertices so that each (Subscript[s, i]) divides the subsequent (Subscript[s, i+1]).

Here are some examples:

If n=3, then the vertex set is V(3)={2,3}. The simplicial complex is K(3)={{2},{3}}. There are no 1-dimensional simplices since 2 does not divide 3, so {2,3} is excluded.

If n=6, then the vertex set is V(6)={2,3,5,6}. The simplicial complex is K(6)={{2},{3},{5},{6},{2,6},{3,6}}.

If n=10, then the vertex set is V(10)={2,3,5,6,7,10}. The simplicial complex is K(10)={{2},{3},{5},{6},{7},{10},{2,6},{2,10},{3,6},{5,10}}.

Write a function that computes this simplicial complex for given n. The input and output of your function should be as follows:
Input: A positive integer n.
Output: The simplicial complex K(n) as defined above.