# Overview of the simplextree package

A simplicial complex S is a pair S = (V, \Sigma) where V is a vertex set and \Sigma a collection of simplices s \in \Sigma satisfying:

1. If v \in V, then v \in S
2. If \tau \subset \sigma and \sigma \in S, then \tau \in S

A simplicial complex is a natural generalization of a graph—any graph can also be represented by a simplicial complex (though the converse is not true!).

Like graphs, there are many ways to represent simplicial complexes in memory. One such way is to use a Simplex Tree: an ordered, trie-like structure whose nodes are in bijection with the faces of the complex. Here’s a picture of a simplicial 3-complex (left) and its corresponding Simplex Tree (right):

To construct the complex above with a simplextree package, simply give the maximal simplices:

from simplextree import SimplexTree
st = SimplexTree([[1,2,3],[2,3,4,5],[6,7,9],[7,8],[10]]) # complex form the picture
print(st)
Simplex Tree with (10, 12, 6, 1) (0, 1, 2, 3)-simplices

To look at the tree structure, use print_tree (see also: print_cousins)

st.print_tree()
1 (h = 2): .( 2 3 )..( 3 )
2 (h = 3): .( 3 4 5 )..( 4 5 5 )...( 5 )
3 (h = 2): .( 4 5 )..( 5 )
4 (h = 1): .( 5 )
5 (h = 0):
6 (h = 2): .( 7 9 )..( 9 )
7 (h = 1): .( 8 9 )
8 (h = 0):
9 (h = 0):
10 (h = 0): 

To maintain fast coface lookup and enumeration, extra links are added between nodes at the same level, which are shown by the dash-dotted lines in the figure above (only for the nodes with the label 5). To view these links with the SimplexTree class, use print_cousins():

st.print_cousins()
(last=2, depth=2): { 1 2 }
(last=3, depth=2): { 1 3 } { 2 3 }
(last=4, depth=2): { 2 4 } { 3 4 }
(last=5, depth=2): { 2 5 } { 3 5 } { 4 5 }
(last=7, depth=2): { 6 7 }
(last=8, depth=2): { 7 8 }
(last=9, depth=2): { 6 9 } { 7 9 }
(last=3, depth=3): { 1 2 3 }
(last=4, depth=3): { 2 3 4 }
(last=5, depth=3): { 2 3 5 } { 2 4 5 } { 3 4 5 }
(last=9, depth=3): { 6 7 9 }
(last=5, depth=4): { 2 3 4 5 }