Veering triangulation demo

Authors:

  • Vincent Delecroix

  • Saul Schleimer

License:

CC BY-NC-SA 3.0

  • 27 Aug. 2018 Toronto

  • 20 Sept. 2018 Temple

  • 25 Sept. 2018 Villetaneuse and CUNY

veerer is a Python and SageMath library for exploration of veering triangulations. It is written by Mark Bell, Vincent Delecroix and Saul Schleimer. It is part of a project that also involve Vaibhav Gadre and Rodolfo Gutiérrez-Romo, see arXiv:1909.00890 [math.DS].

veerer works in conjunction with

  • the Python library pplpy (rational polytope computations and linear optimiation). Note that it is shipped with SageMath (see below).

  • The Python library PyNormaliz (polytope, in particular over number fields).

  • The software SageMath (plotting, linear algebra and many other things)

  • The SageMath library surface_dynamics (for analyzing stratum components)

To import all features from veerer one usually starts with the following lines:

sage: from veerer import *  # random output due to deprecation warnings from realalg
sage: from surface_dynamics import *   # optional - surface_dynamics

To input a triangulation in the program one needs to specify a list of triangles (i, j, k) and the pairing of edges is by convention given by i <-> ~i. The veering coloring is then specified by a list of colors.

sage: # standard torus made of two triangles
sage: faces0 = '(0,1,2)(~0,~1,~2)'
sage: colours0 = 'RRB'
sage: T0 = VeeringTriangulation(faces0, colours0)

sage: T0.genus()
1

sage: T0.angles()
[2]
sage: T0.stratum()  # optional - surface_dynamics
H_1(0)

sage: # triangulation is core if it carries some flat structure
sage: T0.is_core()
True

sage: FS0 = T0.flat_structure_min()
sage: FS0.plot().show(figsize=5)

sage: FFS0 = T0.flat_structure_geometric_middle()
sage: FFS0.plot().show(figsize=5)

sage: # an Abelian genus 2 example
sage: faces1 = '(0, ~3, 4)(1, 2, ~7)(3, ~1, ~2)(5, ~8, ~4)(6, ~5, 8)(7, ~6, ~0)'
sage: colours1 = 'RBRRBRBRB'
sage: T1 = VeeringTriangulation(faces1, colours1)

sage: T1.angles()
[6]
sage: T1.stratum()  # optional - surface_dynamics
H_2(2)

sage: FS1 = T1.flat_structure_min()
sage: FS1.plot().show(figsize=5)

sage: FFS1 = T1.flat_structure_geometric_middle()
sage: FFS1.plot().show(figsize=5)

sage: # a quadratic genus 1 example
sage: faces2 = '(0,1,2)(~0,~3,~8)(3,5,4)(~4,~1,~5)(6,7,8)(~6,9,~2)'
sage: colours2 = 'BRBRBBBRBR'
sage: T2 = VeeringTriangulation(faces2, colours2)

sage: T2.angles()
[3, 3, 1, 1]
sage: T2.stratum()  # optional - surface_dynamics
Q_1(1^2, -1^2)

sage: FS2 = T2.flat_structure_min()
sage: FS2.plot().show(figsize=5)  # not tested (warning from matplotlib)

Viewing train-tracks!

Recall that a veering triangulation is just a pair of transversal train-tracks.

sage: TT_horiz = FS1.plot(horizontal_train_track=True, edge_labels=False)
sage: TT_vert = FS1.plot(vertical_train_track=True, edge_labels=False)
sage: graphics_array([TT_horiz, TT_vert], 1, 2).show(figsize=6)

sage: # constructing veering triangulations from a component stratum
sage: Q = Stratum([3,3,3,3], 2)     # optional - surface_dynamics
sage: Qreg = Q.regular_component()      # optional - surface_dynamics
sage: Qirr = Q.irregular_component()    # optional - surface_dynamics

sage: vt = VeeringTriangulation.from_stratum(Qreg)  # optional - surface_dynamics
sage: vt.stratum()                                  # optional - surface_dynamics
Q_4(3^4)

sage: vt = VeeringTriangulation.from_stratum(Qirr)  # optional - surface_dynamics
sage: vt.stratum()                                  # optional - surface_dynamics
Q_4(3^4)

sage: # constructing a veering triangulation from a pseudo-Anosov homeomorphism
sage: import flipper                    # optional - flipper
sage: S_2_1 = flipper.load('S_2_1')     # optional - flipper
sage: h = S_2_1.mapping_class('abcD')   # optional - flipper
sage: print(h.nielsen_thurston_type())  # optional - flipper
Pseudo-Anosov

sage: VeeringTriangulation.from_pseudo_anosov(h)  # optional - flipper
VeeringTriangulation("(0,~3,~1)(1,2,14)(3,~5,~13)(4,~12,~8)(5,6,~11)(7,8,13)(9,~6,~7)(10,~0,11)(12,~14,~10)(~9,~4,~2)", "RBRBRRBRBBBBRBR")

Core vs not core

sage: # start from our surface in H(2) and let us flip some edges
sage: S = T1.copy(mutable=True)
sage: print(S.is_core())
True
sage: print(S.flippable_edges())
[0, 2, 3, 7, 8]

sage: S.flip(3, BLUE)
sage: print(S.is_core())
True
sage: print(S.flippable_edges())
[3, 7, 8]

sage: S.flip(8, BLUE)
sage: print(S.is_core())
True
sage: print(S.flippable_edges())
[3, 4, 7, 8]

sage: S.flip(4, RED)
sage: print(S.is_core())
True
sage: print(S.flippable_edges())
[4, 7]

sage: FS = S.flat_structure_min()
sage: FS.plot()
Graphics object consisting of 37 graphics primitives

sage: # in the geometric setting, the flipped edge is forced to be BLUE
sage: S.flip(7, RED)
sage: S.is_core()
False

sage: print(S.train_track_polytope(HORIZONTAL))
Cone of dimension 4 in ambient dimension 9 made of 5 facets (backend=ppl)
sage: print(S.train_track_polytope(VERTICAL))
Cone of dimension 3 in ambient dimension 9 made of 3 facets (backend=ppl)

sage: # check that we indeed started with a core veering triangulation
sage: print(T1.train_track_polytope(HORIZONTAL))
Cone of dimension 4 in ambient dimension 9 made of 4 facets (backend=ppl)
sage: print(T1.train_track_polytope(VERTICAL))
Cone of dimension 4 in ambient dimension 9 made of 5 facets (backend=ppl)

Geometric polytope

A triangulation is geometric if it is the L^infinity-Delaunay triangulation of some flat structure

sage: # triangulation of some flat structure
sage: T0.is_geometric()
True

The geometric polytope that parametrizes the geometric vectors is a sub-polytope of the product of the two train-track polytopes.

sage: print(T1.is_geometric())
True
sage: print(T1.geometric_polytope())
Cone of dimension 8 in ambient dimension 18 made of 13 facets (backend=ppl)

Core automaton

The core automaton of a given triangulations T_0 is the directed graph whose vertices are core veering triangulations that can be reached from T_0 by a sequence of flips and there is a directed edge T_i to T_j if T_j is obtained from T_i by a flip.

sage: # T0 was the torus example
sage: from veerer import CoreAutomaton
sage: A0 = CoreAutomaton(T0)
sage: A0
Core veering automaton with 2 vertices

sage: print(A0.num_states(), A0.num_transitions())
2 4
sage: print(sum(vt.is_geometric() for vt in A0))
2
sage: print(sum(vt.is_cylindrical() for vt in A0))
2

sage: # T1 was the genus 2 example in H(2)
sage: A1 = CoreAutomaton(T1)

sage: print(A1.num_states(), A1.num_transitions())
86 300
sage: print(sum(vt.is_geometric() for vt in A1))
54
sage: print(sum(vt.is_cylindrical() for vt in A1))
24

sage: # T2 was the genus 1 example in Q(1^2, -1^2)
sage: A2 = CoreAutomaton(T2)
sage: print(A2.num_states(), A2.num_transitions())
1074 3620
sage: print(sum(vt.is_geometric() for vt in A2))
270
sage: print(sum(vt.is_cylindrical() for vt in A2))
196

Some data (orientable case)

component

dim

core

geometric

cylindrical

H(0)

2

2

2

2

H(2)

4

86

54

24

H(1,1)

5

876

396

136

H(4)^hyp

6

9116

2916

636

H(4)^odd

6

47552

35476

1970

H(2,2)^hyp

7

111732

24192

3934

H(2,2)^odd

7

874750

711568

12740

H(3,1)

7

2011366

1317136

33164

To give an idea about the complexity and timings when generating the above data, here are the steps involved. The timings are for the stratum component H(4)^hyp that is the fourth row in the above array: - generating the core graph ~20 secs for H(4)^hyp (the graph has 9116 vertices and 44664 edges) - filtering the geometric triangulations (single test involves a polytope computation) ~20 secs for H(4)^hyp - filtering cylindrical (single test is cheap) ~2 sec for H(4)^hyp

sage: H = Stratum([4], 1).hyperelliptic_component()  # optional - surface_dynamics
sage: V = VeeringTriangulation.from_stratum(H)         # optional - surface_dynamics
sage: AV = CoreAutomaton(V)                          # long time - ~21 secs # optional - surface_dynamics
sage: print(AV.num_states())                         # long time - ~150 µs # optional - surface_dynamics
9116
sage: sum(v.is_geometric() for v in AV)              # long time - ~21 secs # optional - surface_dynamics
2916
sage: sum(v.is_cylindrical() for v in AV)            # long time - ~1.5 secs # optional - surface_dynamics
636

License

This document is published under the Creative Commons CC BY-SA 3.0.