twist_space

twist_space#

Homology of cylinder decomposition.

class surface_dynamics.flat_surfaces.twist_space.TwistSpace(cd)#

Subspace of relative homology generated by horizontal saddle connections.

This subspace contains the core curves of cylinders. The subspace spanned by the latter is an isotropic subspace of absolute homology.

EXAMPLES:

sage: from surface_dynamics import CylinderDiagram
sage: from surface_dynamics.flat_surfaces.twist_space import TwistSpace
sage: cd = CylinderDiagram("(0,1)-(0,5) (2)-(4) (3,4)-(1) (5)-(2,3)")
sage: tw = TwistSpace(cd)
sage: tw
TwistSpace('(0,1)-(0,5) (2)-(4) (3,4)-(1) (5)-(2,3)')
sage: tw.homologous_cylinders()
[[2, 3]]
sage: tw.cylinder_dimension()
3

In this example, we get that the last two cylinders (3,4)-(1) and (5)-(2,3) are homologous.

cylinder_core_curves()#

Iterate through the core curves of cylinders as homology elements.

EXAMPLES:

sage: from surface_dynamics.flat_surfaces.separatrix_diagram import CylinderDiagram
sage: from surface_dynamics.flat_surfaces.twist_space import TwistSpace
sage: cd = CylinderDiagram('(0,7,1,2)-(3,6,4,5) (3,6,4,5)-(0,7,1,2)')
sage: tw = TwistSpace(cd)
sage: list(tw.cylinder_core_curves())
[(0, 0, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 1, 0)]

sage: cd = CylinderDiagram('(0,7,3,4)-(0,5,3,6) (1,5,2,6)-(1,7,2,4)')
sage: tw = TwistSpace(cd)
sage: list(tw.cylinder_core_curves())
[(1, 0, 0, 1, 1, 1, 0), (0, 1, 1, 0, 1, 1, 0)]
cylinder_dimension()#

Return the dimension of the span of core curves.

See also the method homological_dimension_of_cylinders for CylinderDiagram.

EXAMPLES:

sage: from surface_dynamics import Stratum
sage: from surface_dynamics.flat_surfaces.twist_space import TwistSpace
sage: for cd in Stratum([1,1,1,1], k=1).cylinder_diagrams():
....:     assert TwistSpace(cd).cylinder_dimension() == cd.homological_dimension_of_cylinders()
homologous_cylinders()#

Return the list of homologous cylinders.

OUTPUT: a list of lists. Each sublist is an equivalence class of > 1 homologous cylinders.

EXAMPLES:

sage: from surface_dynamics import Stratum
sage: from surface_dynamics.flat_surfaces.twist_space import TwistSpace
sage: for cd in Stratum([1,1,1,1], k=1).cylinder_diagrams(2):
....:     hom_cyls = TwistSpace(cd).homologous_cylinders()
....:     if hom_cyls:
....:         print(cd)
....:         print(hom_cyls)
(0,7,1,2)-(3,6,4,5) (3,6,4,5)-(0,7,1,2)
[[0, 1]]
Contents