The space of Abelian (or quadratic) differentials is stratified by the degrees of the zeroes (and simple poles for quadratic differentials). Each stratum has one, two or three connected components and each is associated to an (extended) Rauzy class. The connected_components() method (only available for Abelian stratum) give the decomposition of a stratum (which corresponds to the SAGE object AbelianStratum).
The work for Abelian differentials was done by Maxim Kontsevich and Anton Zorich in [KonZor03] and for quadratic differentials by Erwan Lanneau in [Lan08]. Zorich gave an algorithm to pass from a connected component of a stratum to the associated Rauzy class (for both interval exchange transformations and linear involutions) in [Zor08] and is implemented for Abelian stratum at different level (approximately one for each component):
The inverse operation (pass to an interval exchange transformation to the connected component) is partially written in [KonZor03] and simply names here connected_component().
All the code here was first available on Mathematica [ZS].
REFERENCES:
| [KonZor03] | (1, 2) M. Kontsevich, A. Zorich “Connected components of the moduli space of Abelian differentials with prescripebd singularities” Invent. math. 153, 631-678 (2003) |
| [Lan08] | E. Lanneau “Connected components of the strata of the moduli spaces of quadratic differentials”, Annales sci. de l’ENS, serie 4, fascicule 1, 41, 1-56 (2008) |
| [Zor08] | (1, 2, 3, 4, 5) A. Zorich “Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials”, Journal of Modern Dynamics, vol. 2, no 1, 139-185 (2008) (http://www.math.psu.edu/jmd) |
| [ZS] | Anton Zorich, “Generalized Permutation software” (http://perso.univ-rennes1.fr/anton.zorich/Software/software_en.html) |
NOTE:
The quadratic strata are not yet implemented.
AUTHORS:
EXAMPLES:
Construction of a stratum from a list of singularity degrees:
sage: a = AbelianStratum(1,1)
sage: print a
H(1, 1)
sage: print a.genus()
2
sage: print a.nintervals()
5
sage: a = AbelianStratum(4,3,2,1)
sage: print a
H(4, 3, 2, 1)
sage: print a.genus()
6
sage: print a.nintervals()
15
By convention, the degrees are always written in decreasing order:
sage: a1 = AbelianStratum(4,3,2,1)
sage: a1
H(4, 3, 2, 1)
sage: a2 = AbelianStratum(2,3,1,4)
sage: a2
H(4, 3, 2, 1)
sage: a1 == a2
True
It’s also possible to consider stratum with an incoming or an outgoing separtrix marked (the aim of this consideration is to attached a specified degree at the left or the right or the associated interval exchange transformation):
sage: a_out = AbelianStratum(1, 1, marked_separatrix='out')
sage: a_out
H^out(1, 1)
sage: a_in = AbelianStratum(1, 1, marked_separatrix='in')
sage: a_in
H^in(1, 1)
sage: a_out == a_in
False
Get a list of strata with constraints on genus or on the number of intervals of a representative:
sage: for a in AbelianStrata(genus=3):
... print a
H(4)
H(3, 1)
H(2, 2)
H(2, 1, 1)
H(1, 1, 1, 1)
sage: for a in AbelianStrata(nintervals=5):
... print a
H^out(0, 2)
H^out(2, 0)
H^out(1, 1)
H^out(0, 0, 0, 0)
sage: for a in AbelianStrata(genus=2, nintervals=5):
... print a
H^out(0, 2)
H^out(2, 0)
H^out(1, 1)
Obtains the connected components of a stratum:
sage: a = AbelianStratum(0)
sage: print a.connected_components()
[H_hyp(0)]
sage: a = AbelianStratum(6)
sage: cc = a.connected_components()
sage: print cc
[H_hyp(6), H_odd(6), H_even(6)]
sage: for c in cc:
... print c, "\n", c.representative(alphabet=range(1,9))
H_hyp(6)
1 2 3 4 5 6 7 8
8 7 6 5 4 3 2 1
H_odd(6)
1 2 3 4 5 6 7 8
4 3 6 5 8 7 2 1
H_even(6)
1 2 3 4 5 6 7 8
6 5 4 3 8 7 2 1
sage: a = AbelianStratum(1, 1, 1, 1)
sage: print a.connected_components()
[H_c(1, 1, 1, 1)]
sage: c = a.connected_components()[0]
sage: print c.representative(alphabet="abcdefghi")
a b c d e f g h i
e d c f i h g b a
The zero attached on the left of the associated Abelian permutation corresponds to the first singularity degree:
sage: a = AbelianStratum(4, 2, marked_separatrix='out')
sage: b = AbelianStratum(2, 4, marked_separatrix='out')
sage: print a == b
False
sage: print a, ":", a.connected_components()
H^out(4, 2) : [H_odd^out(4, 2), H_even^out(4, 2)]
sage: print b, ":", b.connected_components()
H^out(2, 4) : [H_odd^out(2, 4), H_even^out(2, 4)]
sage: a_odd, a_even = a.connected_components()
sage: b_odd, b_even = b.connected_components()
The representatives are hence different:
sage: print a_odd.representative(alphabet=range(1,10))
1 2 3 4 5 6 7 8 9
4 3 6 5 7 9 8 2 1
sage: print b_odd.representative(alphabet=range(1,10))
1 2 3 4 5 6 7 8 9
4 3 5 7 6 9 8 2 1
sage: print a_even.representative(alphabet=range(1,10))
1 2 3 4 5 6 7 8 9
6 5 4 3 7 9 8 2 1
sage: print b_even.representative(alphabet=range(1,10))
1 2 3 4 5 6 7 8 9
7 6 5 4 3 9 8 2 1
You can retrieve the decomposition of the irreducible Abelian permutations into Rauzy diagrams from the classification of strata:
sage: a = AbelianStrata(nintervals=4)
sage: l = sum([stratum.connected_components() for stratum in a], [])
sage: n = map(lambda x: x.rauzy_diagram().cardinality(), l)
sage: for c,i in zip(l,n):
... print c, ":", i
H_hyp^out(2) : 7
H_hyp^out(0, 0, 0) : 6
sage: print sum(n)
13
sage: a = AbelianStrata(nintervals=5)
sage: l = sum([stratum.connected_components() for stratum in a], [])
sage: n = map(lambda x: x.rauzy_diagram().cardinality(), l)
sage: for c,i in zip(l,n):
... print c, ":", i
H_hyp^out(0, 2) : 11
H_hyp^out(2, 0) : 35
H_hyp^out(1, 1) : 15
H_hyp^out(0, 0, 0, 0) : 10
sage: print sum(n)
71
sage: a = AbelianStrata(nintervals=6)
sage: l = sum([stratum.connected_components() for stratum in a], [])
sage: n = map(lambda x: x.rauzy_diagram().cardinality(), l)
sage: for c,i in zip(l,n):
... print c, ":", i
H_hyp^out(4) : 31
H_odd^out(4) : 134
H_hyp^out(0, 2, 0) : 66
H_hyp^out(2, 0, 0) : 105
H_hyp^out(0, 1, 1) : 20
H_hyp^out(1, 1, 0) : 90
H_hyp^out(0, 0, 0, 0, 0) : 15
sage: print sum(n)
461
Abelian strata.
INPUT:
EXAMPLES:
Abelian strata with a given genus:
sage: for s in AbelianStrata(genus=1): print s
H(0)
sage: for s in AbelianStrata(genus=2): print s
H(2)
H(1, 1)
sage: for s in AbelianStrata(genus=3): print s
H(4)
H(3, 1)
H(2, 2)
H(2, 1, 1)
H(1, 1, 1, 1)
sage: for s in AbelianStrata(genus=4): print s
H(6)
H(5, 1)
H(4, 2)
H(4, 1, 1)
H(3, 3)
H(3, 2, 1)
H(3, 1, 1, 1)
H(2, 2, 2)
H(2, 2, 1, 1)
H(2, 1, 1, 1, 1)
H(1, 1, 1, 1, 1, 1)
Abelian strata with a given number of intervals:
sage: for s in AbelianStrata(nintervals=2): print s
H^out(0)
sage: for s in AbelianStrata(nintervals=3): print s
H^out(0, 0)
sage: for s in AbelianStrata(nintervals=4): print s
H^out(2)
H^out(0, 0, 0)
sage: for s in AbelianStrata(nintervals=5): print s
H^out(0, 2)
H^out(2, 0)
H^out(1, 1)
H^out(0, 0, 0, 0)
Abelian strata with both constraints:
sage: for s in AbelianStrata(genus=2, nintervals=4): print s
H^out(2)
sage: for s in AbelianStrata(genus=5, nintervals=12): print s
H^out(8, 0, 0)
H^out(0, 8, 0)
H^out(0, 7, 1)
H^out(1, 7, 0)
H^out(7, 1, 0)
H^out(0, 6, 2)
H^out(2, 6, 0)
H^out(6, 2, 0)
H^out(1, 6, 1)
H^out(6, 1, 1)
H^out(0, 5, 3)
H^out(3, 5, 0)
H^out(5, 3, 0)
H^out(1, 5, 2)
H^out(2, 5, 1)
H^out(5, 2, 1)
H^out(0, 4, 4)
H^out(4, 4, 0)
H^out(1, 4, 3)
H^out(3, 4, 1)
H^out(4, 3, 1)
H^out(2, 4, 2)
H^out(4, 2, 2)
H^out(2, 3, 3)
H^out(3, 3, 2)
Bases: sage.combinat.combinat.InfiniteAbstractCombinatorialClass
Abelian strata.
Bases: sage.combinat.combinat.CombinatorialClass
Strata with constraint number of intervals.
INPUT:
Bases: sage.combinat.combinat.CombinatorialClass
Stratas of genus g surfaces.
INPUT:
Bases: sage.combinat.combinat.CombinatorialClass
Abelian strata of presrcribed genus and number of intervals.
INPUT:
Bases: sage.structure.sage_object.SageObject
Stratum of Abelian differentials.
A stratum with a marked outgoing separatrix corresponds to Rauzy diagram with left induction, a stratum with marked incoming separatrix correspond to Rauzy diagram with right induction. If there is no marked separatrix, the associated Rauzy diagram is the extended Rauzy diagram (consideration of the sage.combinat.iet.template.Permutation.symmetric() operation of Boissy-Lanneau).
When you want to specify a marked separatrix, the degree on which it is is the first term of your degrees list.
INPUT:
EXAMPLES:
Creation of an Abelian stratum and get it’s connected components:
sage: a = AbelianStratum(2, 2)
sage: print a
H(2, 2)
sage: a.connected_components()
[H_hyp(2, 2), H_odd(2, 2)]
Specification of marked separatrix:
sage: a = AbelianStratum(4,2,marked_separatrix='in')
sage: print a
H^in(4, 2)
sage: b = AbelianStratum(2,4,marked_separatrix='in')
sage: print b
H^in(2, 4)
sage: a == b
False
sage: a = AbelianStratum(4,2,marked_separatrix='out')
sage: print a
H^out(4, 2)
sage: b = AbelianStratum(2,4,marked_separatrix='out')
sage: print b
H^out(2, 4)
sage: a == b
False
Get a representative of a connected component:
sage: a = AbelianStratum(2,2)
sage: a_hyp, a_odd = a.connected_components()
sage: print a_hyp.representative()
1 2 3 4 5 6 7
7 6 5 4 3 2 1
sage: print a_odd.representative()
0 1 2 3 4 5 6
3 2 4 6 5 1 0
You can precise the alphabet:
sage: print a_odd.representative(alphabet="ABCDEFGHIJKLMNOPQRSTUVWXYZ")
A B C D E F G
D C E G F B A
By default, you get a reduced permutation, but you can specify that you want a labelled one:
sage: p_reduced = a_odd.representative()
sage: p_labelled = a_odd.representative(reduced=False)
Lists the connected components of the Stratum.
OUTPUT:
list – a list of connected components of stratum
EXAMPLES:
sage: AbelianStratum(0).connected_components()
[H_hyp(0)]
sage: AbelianStratum(2).connected_components()
[H_hyp(2)]
sage: AbelianStratum(1,1).connected_components()
[H_hyp(1, 1)]
sage: AbelianStratum(4).connected_components()
[H_hyp(4), H_odd(4)]
sage: AbelianStratum(3,1).connected_components()
[H_c(3, 1)]
sage: AbelianStratum(2,2).connected_components()
[H_hyp(2, 2), H_odd(2, 2)]
sage: AbelianStratum(2,1,1).connected_components()
[H_c(2, 1, 1)]
sage: AbelianStratum(1,1,1,1).connected_components()
[H_c(1, 1, 1, 1)]
Returns the genus of the stratum.
OUTPUT:
integer – the genus
EXAMPLES:
sage: AbelianStratum(0).genus()
1
sage: AbelianStratum(1,1).genus()
2
sage: AbelianStratum(3,2,1).genus()
4
Tests if the strata is connected.
OUTPUT:
boolean – True if it’s connected else False
EXAMPLES:
sage: AbelianStratum(2).is_connected()
True
sage: AbelianStratum(2).connected_components()
[H_hyp(2)]
sage: AbelianStratum(2,2).is_connected()
False
sage: AbelianStratum(2,2).connected_components()
[H_hyp(2, 2), H_odd(2, 2)]
Returns the number of intervals of any iet of the strata.
OUTPUT:
integer – the number of intervals for any associated iet
EXAMPLES:
sage: AbelianStratum(0).nintervals()
2
sage: AbelianStratum(0,0).nintervals()
3
sage: AbelianStratum(2).nintervals()
4
sage: AbelianStratum(1,1).nintervals()
5
Bases: sage.structure.sage_object.SageObject
Connected component of Abelian stratum.
Warning
Internal class! Do not use directly!
TESTS:
Tests for outgoing marked separatrices:
sage: a = AbelianStratum(4,2,0,marked_separatrix='out')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_out_degree()
4
sage: a_even.representative().attached_out_degree()
4
sage: a = AbelianStratum(2,4,0,marked_separatrix='out')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_out_degree()
2
sage: a_even.representative().attached_out_degree()
2
sage: a = AbelianStratum(0,4,2,marked_separatrix='out')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_out_degree()
0
sage: a_even.representative().attached_out_degree()
0
sage: a = AbelianStratum(3,2,1,marked_separatrix='out')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_out_degree()
3
sage: a = AbelianStratum(2,3,1,marked_separatrix='out')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_out_degree()
2
sage: a = AbelianStratum(1,3,2,marked_separatrix='out')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_out_degree()
1
Tests for incoming separatrices:
sage: a = AbelianStratum(4,2,0,marked_separatrix='in')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_in_degree()
4
sage: a_even.representative().attached_in_degree()
4
sage: a = AbelianStratum(2,4,0,marked_separatrix='in')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_in_degree()
2
sage: a_even.representative().attached_in_degree()
2
sage: a = AbelianStratum(0,4,2,marked_separatrix='in')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_in_degree()
0
sage: a_even.representative().attached_in_degree()
0
sage: a = AbelianStratum(3,2,1,marked_separatrix='in')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_in_degree()
3
sage: a = AbelianStratum(2,3,1,marked_separatrix='in')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_in_degree()
2
sage: a = AbelianStratum(1,3,2,marked_separatrix='in')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_in_degree()
1
Returns the genus of the surfaces in this connected component.
OUTPUT:
integer – the genus of the surface
EXAMPLES:
sage: a = AbelianStratum(6,4,2,0,0)
sage: c_odd, c_even = a.connected_components()
sage: c_odd.genus()
7
sage: c_even.genus()
7
sage: a = AbelianStratum([1]*8)
sage: c = a.connected_components()[0]
sage: c.genus()
5
Returns the number of intervals of the representative.
OUTPUT:
integer – the number of intervals in any representative
EXAMPLES:
sage: a = AbelianStratum(6,4,2,0,0)
sage: c_odd, c_even = a.connected_components()
sage: c_odd.nintervals()
18
sage: c_even.nintervals()
18
sage: a = AbelianStratum([1]*8)
sage: c = a.connected_components()[0]
sage: c.nintervals()
17
The stratum of this component
OUTPUT:
stratum - the stratum where this component leaves
EXAMPLES:
sage: p = iet.Permutation('a b','b a')
sage: c = p.connected_component()
sage: c.parent()
H(0)
Returns the Rauzy diagram associated to this connected component.
OUTPUT:
rauzy diagram – the Rauzy diagram associated to this stratum
EXAMPLES:
sage: c = AbelianStratum(0).connected_components()[0]
sage: r = c.rauzy_diagram()
Returns the Zorich representative of this connected component.
Zorich constructs explcitely interval exchange transformations for each stratum in [Zor08].
INPUT:
OUTPUT:
permutation – a permutation which lives in this component
EXAMPLES:
sage: c = AbelianStratum(1,1,1,1).connected_components()[0]
sage: print c
H_c(1, 1, 1, 1)
sage: p = c.representative(alphabet=range(9))
sage: print p
0 1 2 3 4 5 6 7 8
4 3 2 5 8 7 6 1 0
sage: p.connected_component()
H_c(1, 1, 1, 1)
Bases: sage.combinat.iet.strata.ConnectedComponentOfAbelianStratum
Connected component of Abelian stratum with even spin structure.
Warning
Internal class! Do not use directly!
Returns the Zorich representative of this connected component.
Zorich constructs explcitely interval exchange transformations for each stratum in [Zor08].
EXAMPLES:
sage: c = AbelianStratum(6).connected_components()[2]
sage: c
H_even(6)
sage: p = c.representative(alphabet=range(8))
sage: p
0 1 2 3 4 5 6 7
5 4 3 2 7 6 1 0
sage: p.connected_component()
H_even(6)
sage: c = AbelianStratum(4,4).connected_components()[2]
sage: c
H_even(4, 4)
sage: p = c.representative(alphabet=range(11))
sage: p
0 1 2 3 4 5 6 7 8 9 10
5 4 3 2 6 8 7 10 9 1 0
sage: p.connected_component()
H_even(4, 4)
Bases: sage.combinat.iet.strata.ConnectedComponentOfAbelianStratum
Hyperelliptic component of Abelian stratum.
Warning
Internal class! Do not use directly!
Returns the Zorich representative of this connected component.
Zorich constructs explcitely interval exchange transformations for each stratum in [Zor08].
INPUT:
EXAMPLES:
sage: c = AbelianStratum(0).connected_components()[0]
sage: c
H_hyp(0)
sage: p = c.representative(alphabet="01")
sage: p
0 1
1 0
sage: p.connected_component()
H_hyp(0)
sage: c = AbelianStratum(0,0).connected_components()[0]
sage: c
H_hyp(0, 0)
sage: p = c.representative(alphabet="abc")
sage: p
a b c
c b a
sage: p.connected_component()
H_hyp(0, 0)
sage: c = AbelianStratum(2).connected_components()[0]
sage: c
H_hyp(2)
sage: p = c.representative(alphabet="ABCD")
sage: p
A B C D
D C B A
sage: p.connected_component()
H_hyp(2)
sage: c = AbelianStratum(1,1).connected_components()[0]
sage: c
H_hyp(1, 1)
sage: p = c.representative(alphabet="01234")
sage: p
0 1 2 3 4
4 3 2 1 0
sage: p.connected_component()
H_hyp(1, 1)
Bases: sage.combinat.iet.strata.ConnectedComponentOfAbelianStratum
Non hyperelliptic component of Abelian stratum.
Warning
Internal class! Do not use directly!
Bases: sage.combinat.iet.strata.ConnectedComponentOfAbelianStratum
Connected component of an Abelian stratum wit odd spin parity.
Warning
Internal class! Do not use directly!
Returns the Zorich representative of this connected component.
Zorich constructs explcitely interval exchange transformations for each stratum in [Zor08].
EXAMPLES:
sage: a = AbelianStratum(6).connected_components()[1]
sage: print a.representative(alphabet=range(8))
0 1 2 3 4 5 6 7
3 2 5 4 7 6 1 0
sage: a = AbelianStratum(4,4).connected_components()[1]
sage: print a.representative(alphabet=range(11))
0 1 2 3 4 5 6 7 8 9 10
3 2 5 4 6 8 7 10 9 1 0
Bases: sage.structure.sage_object.SageObject
Stratum of quadratic differentials.
Returns the genus.
EXAMPLES:
sage: QuadraticStratum(-1,-1,-1,-1).genus()
0