Navigating and querying a complex

Order

Any complex will have simplices of a maximum order:

maxk = c.maxOrder()

Given the constraints on a simplicial complex, a complex with maximum order k will have simplices of all orders from k down to (and including) 0.

It is often handy to be able to query a given simplex for its order:

k = c.orderOf(l45)

It can also be useful to query a complex as to how many simplices it has of each order:

ks = c.numberOfSimplicesOfOrder()

SimplicialComplex.numberOfSimplicesOfOrder() returns a dict mapping orders to the number of simplices of that order. The simplices themselves can be retrieved using SimplicialComplex.simplicesOfOrder().

Retrieving simplices

The simplest opeartion for accessing a complex is to access all the simplex names:

ss = c.simplices()

It is also possible to access only those simplices of a given order, for example:

ss = c.simplicesOfOrder(2)

which retrieves the names of all 2-simplices (triangles). It is sometimes also useful to filter the list of simplices by providing a suitable predicate, mapping a complex and a simplex within it to a boolean:

def atleast(c, s):
    return (c[s]['value'] > 5])

ss = c.allSimplices(atleast)

which would retrieve all simplices having a value attribute greater than 5.

Finding simplices

Testing whether a simplex is in a complex can happen through a procedural or an operator interface:

if c.contrainsSimplex(l23):
    print('yes')
else:
    if l13 in c:
        print('but yes')

Another frequent operation is to check whether the complex contains a simplex with a given basis:

if c.containsSimplexWithBasis([ s1, s2, s3 ]):
    print('yes')

The simplex itself can also be retrieved directly:

s = c.simplexWithBasis([ s1, s2, s3 ])
if s is not None:
    print('we have simplex {s}'.format(s = s))
else:
    print('no')

Traversing the simplex hierarchy

Simplices are arranged in a hierarchy according to the face relationship. A k-simplex must by definition have (k + 1) (km- 1)-simplices as its faces. We can traverse up and down this hierarchy from a startign simplex. To find the faces of a simplex, we use:

fs = c.faces(l23)

SimplicialComplex.faces() is “one-step”. in the sense that it returns the simplices that are faces of this simplex – but not the faces of those simplices, and so on. For this we nee the transitive closure of the operation:

fs = c.closureOf(l23)

which returns all the simplices that make up the given simplex, down to its basis (its closure). We can, if desired, skip straight to the basis:

fs = c.basisOf(l23)

In the opposite direction, to find those simplices a simplex is a face of, we use:

fs = c.faceOf(l23)

This is again a “single-step” operation, and we may also need a transitive closure:

fs = c.partOf(l23)

(In the topology literature this operation if often called the star of the starting simplex.)

Building and navigating a complex are only building blocks to its Analysing a complex using various tools from topology.