Python module for experiments in mapping class groups.
The project is still under construction, but some functionality already exists, see below.
Download the project and run
python setup.py install
from the directory containing the setup.py file. Without root access you can run python setup.py install --user.
The program currently run both under Python 2.7 and Python 3, and requires the Numpy and NetworkX packages. It does not require Sage.
- Defining the Humphries generators on closed surfaces.
>>> from macaw import humphries_generators
>>> A, B, c = humphries_generators(3) # Humphries generators on the genus 3 surface.
- Checking is a mapping class is the identity (i.e., solution to the word problem).
>>> f = A[0]*A[1]
>>> f.is_identity()
False
>>> g = A[0]*A[1]*A[0]**(-1)*A[1]**(-1) # A[0] and A[1] are disjoint curves, so they commute.
>>> g.is_identity()
True
- Checking relations in the mapping class group (as an immediate application of the solution for the word problem).
>>> A[0]*A[1] == A[1]*A[0] # Dehn twist about disjoint curves commute
True
>>> A[0]*B[0] == B[0]*A[0] # Dehn twists about curves intersecting once do not commute.
False
>>> A[0]*B[0]*A[0] == B[0]*A[0]*B[0] # However, they satisfy the braid relation.
True
- Approximating stretch factors (currently in a very dumb way).
>>> f = A[0]*B[0]**(-1) # partial pA supported on a torus
>>> f.stretch_factor()
2.61803398874990
- Computing orders.
>>> f = A[0]*B[0]**(-1)
>>> f.order()
0
>>> from macaw import hyperelliptic_involution
>>> g = hyperelliptic_involution(3) # Hyperelliptic involution on the genus 3 surface.
>>> g.order()
2
- Computing the action on homology.
>>> A, B, c = humphries_generators(2)
>>> f = A[0]*B[0]**(-1)
>>> f.action_on_homology()
[ 2 -1 1 0]
[ 0 1 0 0]
[ 1 -1 1 0]
[ 0 0 0 1]
>>> g = hyperelliptic_involution(2)
>>> g.action_on_homology()
matrix([[-1, 0, 0, 0],
[0, -1, 0, 0],
[0, 0, -1, 0],
[0, 0, 0, -1]], dtype=object)