single_cylinder

Single cylinder permutation representatives.

This file gathers the functions necessary for AbelianStratum.single_cylinder_representative(), AbelianStratumComponent.single_cylinder_representative(), HypAbelianStratumComponent.single_cylinder_representative(), EvenAbelianStratumComponent.single_cylinder_representative(), and OddAbelianStratumComponent.single_cylinder_representative() in abelian_strata.

TESTS:

sage: from surface_dynamics import * sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

Check for non-hyperelliptic strata:

sage: for d in range(2, 12):
....:     for A in AbelianStrata(dimension=d, fake_zeros=True):
....:         for C in A.components():
....:             if C._name == 'hyp': continue
....:             o = C.single_cylinder_origami()
....:             r = o.r().cycle_tuples(singletons=True)
....:             u = o.u().cycle_tuples(singletons=True)
....:             assert len(r) == 1 and len(u) == 1
....:             assert o.stratum_component(True) == C

surface_dynamics.flat_surfaces.single_cylinder.cylinder_concatenation(perm_1, perm_2, alphabet=None)

Combines two single cylinder permutation representatives.

Combines two single cylinder permutation representatives of connected components of Abelian strata to produce another single cylinder representative of a different stratum.

Such a method was described by Jeffreys [Jef19].

INPUT:

• perm_1, perm_2 - two single cylinder permutation representatives.

• alphabet - alphabet or None (default: None): whether you want to specify an alphabet for your representative.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

We first take two single cylinder permutation representatives for the odd components of H_3(4)^odd and H_4(6)^odd.:

sage: perm_1 = Stratum([4], k=1).odd_component().single_cylinder_representative()
sage: perm_1
0 1 2 3 4 5
2 5 4 1 3 0
sage: perm_1.stratum_component() == Stratum([4], k=1).odd_component()
True
sage: _cylinder_check(perm_1)
True
sage: perm_2 = Stratum([6], k=1).odd_component().single_cylinder_representative()
sage: perm_2
0 1 2 3 4 5 6 7
2 5 4 7 3 1 6 0
sage: perm_2.stratum_component() == Stratum([6], k=1).odd_component()
True
sage: _cylinder_check(perm_2)
True

We check that the cylinder_concatenation of these permutations produces a single cylinder permutation representative of the connected component H_6(6,4)^odd.:

sage: perm_3 = cylinder_concatenation(perm_1,perm_2)
sage: perm_3
0 1 2 3 4 5 6 7 8 9 10 11 12
2 5 4 6 3 7 10 9 12 8 1 11 0
sage: perm_3.stratum_component() == Stratum([6,4], k=1).odd_component()
True
sage: _cylinder_check(perm_3)
True

We now instead take the cylinder_concatenation of perm_1 with a single cylinder permutation representative of the connected component H_4(6)^even. We see that the resulting permutation is a single cylinder permutation representative of the connected component H_6(6,4)^even.:

sage: perm_4 = Stratum([6], k=1).even_component().single_cylinder_representative()
sage: perm_5 = cylinder_concatenation(perm_1,perm_4,Alphabet(name='lower'))
sage: perm_4
0 1 2 3 4 5 6 7
2 7 6 5 3 1 4 0
sage: perm_4.stratum_component() == Stratum([6], k=1).even_component()
True
sage: _cylinder_check(perm_4)
True
sage: perm_5
a b c d e f g h i j k l m
c f e g d h m l k i b j a
sage: perm_5.stratum_component() == Stratum([6,4], k=1).even_component()
True
sage: _cylinder_check(perm_5)
True

surface_dynamics.flat_surfaces.single_cylinder.even_ones_odds(odd_zeros, one_count)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in an Abelian stratum with odd order zeros and an even, at least four, number of zeros of order 1.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• odd_zeros - an even length list of odd positive integers an even number of which are equal to one.

• one_count - a positive integer equal to the number of ones in real_zeros.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = even_ones_odds([1,1,1,1],4)
sage: perm
0 1 2 3 4 5 6 7 8
2 6 5 3 1 8 4 7 0
sage: perm.stratum_component() == Stratum([1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = even_ones_odds([1,1,1,1,1,1],6)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 8 1 5 11 7 3 10 6 12 9 4 0
sage: perm.stratum_component() == Stratum([1,1,1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = even_ones_odds([5,3,1,1,1,1],4)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 6 5 3 9 8 4 7 10 13 12 15 11 18 14 17 1 16 0
sage: perm.stratum_component() == Stratum([5,3,1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = even_ones_odds([3,3,1,1,1,1,1,1],6)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 8 13 5 11 7 3 10 6 12 9 4 14 20 18 17 19 16 1 15 0
sage: perm.stratum_component() == Stratum([3,3,1,1,1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.even_twos_even(real_zeros, two_count)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the even component of an Abelian stratum having an even, at least four, number of zeros of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers an even number of which are equal to two.

• two_count - a positive integer equal to the number of twos in real_zeros.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = even_twos_even([2,2,2,2],4)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 5 4 1 12 3 10 7 11 9 6 8 0
sage: perm.stratum_component() == Stratum([2,2,2,2], k=1).even_component()
True
sage: _cylinder_check(perm)
True
sage: perm = even_twos_even([4,2,2,2,2,2,2],6)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
2 5 4 13 12 3 10 7 11 9 6 8 14 16 18 15 19 17 20 23 22 1 21 0
sage: perm.stratum_component() == Stratum({4:1,2:6}, k=1).even_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.even_twos_odd(real_zeros, two_count)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the odd component of an Abelian stratum having an even, at least two, number of zeros of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers an even number of which are equal to two.

• two_count - a positive integer equal to the number of twos in real_zeros.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = even_twos_odd([2,2],2)
sage: perm
0 1 2 3 4 5 6
2 4 6 3 1 5 0
sage: perm.stratum_component() == Stratum([2,2], k=1).odd_component()
True
sage: _cylinder_check(perm)
True
sage: perm = even_twos_odd([4,2,2,2,2],4)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
2 4 6 3 7 5 8 10 12 9 13 11 14 17 16 1 15 0
sage: perm.stratum_component() == Stratum({4:1,2:4}, k=1).odd_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.even_zero_even(num)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the even component of the Abelian stratum with a single zero of the given order.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• num - an even integer at least six.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = even_zero_even(6)
sage: perm
0 1 2 3 4 5 6 7
2 7 6 5 3 1 4 0
sage: perm.stratum_component() == Stratum([6], k=1).even_component()
True
sage: _cylinder_check(perm)
True
sage: perm = even_zero_even(8)
sage: perm
0 1 2 3 4 5 6 7 8 9
2 7 6 5 3 9 4 1 8 0
sage: perm.stratum_component() == Stratum([8], k=1).even_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.even_zero_odd(num)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the odd component of the Abelian stratum with a single zero of the given order.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• num - an even integer at least four.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = even_zero_odd(4)
sage: perm
0 1 2 3 4 5
2 5 4 1 3 0
sage: perm.stratum_component() == Stratum([4], k=1).odd_component()
True
sage: _cylinder_check(perm)
True
sage: perm = even_zero_odd(6)
sage: perm
0 1 2 3 4 5 6 7
2 5 4 7 3 1 6 0
sage: perm.stratum_component() == Stratum([6], k=1).odd_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.min_on_bot(zero_pair)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the Abelian stratum having a pair of zeros of the given odd orders which differ by two.

The permutations have a particular form required for the construction of other representatives.

Such representatives were constructed by Jeffreys [Jef19].

INPUT:

• zero_pair - a list of two odd positive integers at least one and differing by two.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = min_on_bot([3,1])
sage: perm
0 1 2 3 4 5 6
2 6 5 1 4 3 0
sage: perm.stratum_component() == Stratum([3,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = min_on_bot([5,3])
sage: perm
0 1 2 3 4 5 6 7 8 9 10
2 6 4 10 8 3 1 9 7 5 0
sage: perm.stratum_component() == Stratum([5,3], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = min_on_bot([7,5])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 6 4 10 8 3 12 9 7 5 14 11 1 13 0
sage: perm.stratum_component() == Stratum([7,5], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.no_ones_odds(odd_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in an Abelian stratum with odd order zeros and no zeros of order 1.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• odd_zeros - an even length list of odd positive integers none of which are equal to one.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = no_ones_odds([7,3])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 5 4 8 3 7 9 6 12 11 1 10 0
sage: perm.stratum_component() == Stratum([7,3], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = no_ones_odds([5,3,3,3])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 5 4 7 3 10 6 9 11 8 12 18 16 15 17 14 1 13 0
sage: perm.stratum_component() == Stratum([5,3,3,3], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.no_two_even(real_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the even component of an Abelian stratum having no zeros of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers none of which are equal to two.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = no_two_even([4,4])
sage: perm
0 1 2 3 4 5 6 7 8 9 10
2 10 7 5 8 1 9 6 4 3 0
sage: perm.stratum_component() == Stratum([4,4], k=1).even_component()
True
sage: _cylinder_check(perm)
True
sage: perm = no_two_even([6,4])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 7 6 5 3 8 4 9 12 11 1 10 0
sage: perm.stratum_component() == Stratum([6,4], k=1).even_component()
True

surface_dynamics.flat_surfaces.single_cylinder.no_two_odd(real_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the odd component of an Abelian stratum having no zeros of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers none of which are equal to two.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = no_two_odd([6,4])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 5 4 7 3 8 6 9 12 11 1 10 0
sage: perm.stratum_component() == Stratum([6,4], k=1).odd_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.odd_ones_odds(odd_zeros, one_count)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in an Abelian stratum with odd order zeros and an odd, at least five, number of zeros of order 1.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• odd_zeros - an even length list of odd positive integers an odd number of which are equal to one.

• one_count - a positive integer equal to the number of ones in real_zeros.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = odd_ones_odds([5,1,1,1,1,1],5)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 4 7 3 9 8 6 5 10 14 13 11 1 16 12 15 0
sage: perm.stratum_component() == Stratum([5,1,1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odd_ones_odds([3,1,1,1,1,1,1,1],7)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 5 7 6 4 3 8 14 1 11 17 13 9 16 12 18 15 10 0
sage: perm.stratum_component() == Stratum([3,1,1,1,1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.odd_twos_even(real_zeros, two_count)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the even component of an Abelian stratum having an odd, at least three, number of zeros of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers an even number of which are equal to two.

• two_count - a positive integer equal to the number of twos in real_zeros.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = odd_twos_even([2,2,2],3)
sage: perm
0 1 2 3 4 5 6 7 8 9
2 9 8 7 6 3 5 1 4 0
sage: perm.stratum_component() == Stratum([2,2,2], k=1).even_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odd_twos_even([4,2,2,2,2,2],5)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 9 8 7 6 3 5 10 4 11 13 15 12 16 14 17 20 19 1 18 0
sage: perm.stratum_component() == Stratum({4:1,2:5}, k=1).even_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.odd_twos_odd(real_zeros, two_count)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the odd component of an Abelian stratum having an odd, at least three, number of zeros of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers an odd number of which are equal to two.

• two_count - a positive integer equal to the number of twos in real_zeros.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = odd_twos_odd([2,2,2],3)
sage: perm
0 1 2 3 4 5 6 7 8 9
2 8 6 9 4 1 3 5 7 0
sage: perm.stratum_component() == Stratum([2,2,2], k=1).odd_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odd_twos_odd([4,2,2,2,2,2],5)
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 8 6 9 4 10 3 5 7 11 13 15 12 16 14 17 20 19 1 18 0
sage: perm.stratum_component() == Stratum({4:1,2:5}, k=1).odd_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.odd_zeros_one_one(odd_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in an Abelian stratum having odd order zeros of the given orders.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• odd_zeros - an even length list of odd positive integers.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = odd_zeros_one_one([5,5])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 5 4 7 3 9 6 12 8 11 1 10 0
sage: perm.stratum_component() == Stratum([5,5], k=1).non_hyperelliptic_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odd_zeros_one_one([5,1])
sage: perm
0 1 2 3 4 5 6 7 8
2 4 7 3 1 8 6 5 0
sage: perm.stratum_component() == Stratum([5,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odd_zeros_one_one([5,3,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 4 7 3 9 8 6 5 10 13 1 14 12 11 0
sage: perm.stratum_component() == Stratum([5,3,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odd_zeros_one_one([5,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 12 9 8 1 7 3 6 10 5 4 11 0
sage: perm.stratum_component() == Stratum([5,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odd_zeros_one_one([5,3,1,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 6 5 3 9 8 4 7 10 13 12 15 11 18 14 17 1 16 0
sage: perm.stratum_component() == Stratum([5,3,1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odd_zeros_one_one([5,1,1,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 4 7 3 9 8 6 5 10 14 13 11 1 16 12 15 0
sage: perm.stratum_component() == Stratum([5,1,1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.odds_left_swap(zero_pair)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the Abelian stratum having a pair of zeros of the given odd orders.

Performs a column swap on another permutation to achieve this.

Such a method was described by Jeffreys [Jef19].

INPUT:

• zero_pair - a list of two odd positive integers at least three and differing by at least four.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = odds_left_swap([7,3])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 5 4 8 3 7 9 6 12 11 1 10 0
sage: perm.stratum_component() == Stratum([7,3], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odds_left_swap([11,5])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 5 4 7 3 9 6 12 8 11 13 10 16 15 18 14 1 17 0
sage: perm.stratum_component() == Stratum([11,5], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.odds_right_swap(zero_pair)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the Abelian stratum having a pair of zeros of the given odd orders.

Performs a column swap on another permutation to achieve this.

Such a method was described by Jeffreys [Jef19].

INPUT:

• zero_pair - a list of two odd positive integers at least three and differing by zero or two.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = odds_right_swap([3,3])
sage: perm
0 1 2 3 4 5 6 7 8
2 8 6 5 7 4 1 3 0
sage: perm.stratum_component() == Stratum([3,3], k=1).non_hyperelliptic_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odds_right_swap([5,5])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 5 4 7 3 9 6 12 8 11 1 10 0
sage: perm.stratum_component() == Stratum([5,5], k=1).non_hyperelliptic_component()
True
sage: _cylinder_check(perm)
True
sage: perm = odds_right_swap([5,3])
sage: perm
0 1 2 3 4 5 6 7 8 9 10
2 5 4 7 3 10 6 9 1 8 0
sage: perm.stratum_component() == Stratum([5,3], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.one_one_odds(odd_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in an Abelian stratum with odd order zeros and one zero of order 1.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• odd_zeros - an even length list of odd positive integers one of which is equal to one.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = one_one_odds([3,1])
sage: perm
0 1 2 3 4 5 6
2 5 1 6 4 3 0
sage: perm.stratum_component() == Stratum([3,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = one_one_odds([5,1])
sage: perm
0 1 2 3 4 5 6 7 8
2 4 7 3 1 8 6 5 0
sage: perm.stratum_component() == Stratum([5,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = one_one_odds([7,3,3,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 5 4 9 3 8 6 11 10 7 12 18 16 15 17 14 1 13 0
sage: perm.stratum_component() == Stratum([7,3,3,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.one_two_even(real_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the even component of an Abelian stratum having one zero of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers one of which is equal to two.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = one_two_even([4,2])
sage: perm
0 1 2 3 4 5 6 7 8
2 4 1 8 7 5 3 6 0
sage: perm.stratum_component() == Stratum([4,2], k=1).even_component()
True
sage: _cylinder_check(perm)
True
sage: perm = one_two_even([6,2])
sage: perm
0 1 2 3 4 5 6 7 8 9 10
2 10 9 8 6 3 5 1 4 7 0
sage: perm.stratum_component() == Stratum([6,2], k=1).even_component()
True
sage: _cylinder_check(perm)
True
sage: perm = one_two_even([8,6,2])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
2 7 6 5 3 8 4 10 12 9 13 11 14 17 16 19 15 1 18 0
sage: perm.stratum_component() == Stratum([8,6,2], k=1).even_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.one_two_odd(real_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the odd component of an Abelian stratum having one zero of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers one of which is equal to two.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = one_two_odd([4,2])
sage: perm
0 1 2 3 4 5 6 7 8
2 5 8 3 6 4 1 7 0
sage: perm.stratum_component() == Stratum([4,2], k=1).odd_component()
True
sage: _cylinder_check(perm)
True
sage: perm = one_two_odd([8,6,2])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
2 5 4 7 3 8 6 10 12 9 13 11 14 17 16 19 15 1 18 0
sage: perm.stratum_component() == Stratum([8,6,2], k=1).odd_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.only_even_2(odd_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the Abelian stratum having zeros of the given odd orders and a single zero of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• odd_zeros - an even length list of odd positive integers.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = only_even_2([1,1])
sage: perm
0 1 2 3 4 5 6 7
2 6 4 1 7 5 3 0
sage: perm.stratum_component() == Stratum([2,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_even_2([1,1,1,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 6 4 8 7 5 3 9 13 12 10 1 15 11 14 0
sage: perm.stratum_component() == Stratum({2:1,1:6}, k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_even_2([1,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11
2 7 11 6 3 9 5 1 8 4 10 0
sage: perm.stratum_component() == Stratum([2,1,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_even_2([1,1,1,1,1,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
2 7 11 6 3 9 5 12 8 4 10 13 17 16 14 1 19 15 18 0
sage: perm.stratum_component() == Stratum({2:1,1:8}, k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_even_2([3,3,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 6 4 8 7 5 3 9 15 13 12 14 11 1 10 0
sage: perm.stratum_component() == Stratum([3,3,2,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_even_2([3,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13
2 6 4 8 7 5 3 9 12 1 13 11 10 0
sage: perm.stratum_component() == Stratum([3,2,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_even_2([5,3,3,3])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
2 8 6 5 7 4 9 3 11 13 10 14 12 15 21 19 18 20 17 1 16 0
sage: perm.stratum_component() == Stratum([5,3,3,3,2], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.only_odds_11(even_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the Abelian stratum having zeros of the given even orders and two zeros of order one.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• even_zeros - a list of even positive integers.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = only_odds_11([2])
sage: perm
0 1 2 3 4 5 6 7
2 6 4 1 7 5 3 0
sage: perm.stratum_component() == Stratum([2,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_odds_11([2,2])
sage: perm
0 1 2 3 4 5 6 7 8 9 10
2 4 9 7 3 8 5 1 10 6 0
sage: perm.stratum_component() == Stratum([2,2,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_odds_11([4,2])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 6 4 8 7 5 3 9 12 11 1 10 0
sage: perm.stratum_component() == Stratum([4,2,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_odds_11([6])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11
2 6 4 1 8 7 10 9 11 5 3 0
sage: perm.stratum_component() == Stratum([6,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_odds_11([4,4])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 7 4 10 9 5 8 6 3 11 14 13 1 12 0
sage: perm.stratum_component() == Stratum([4,4,1,1],k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = only_odds_11([8,6])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 7 4 14 9 8 11 10 13 5 12 6 3 15 18 17 20 16 1 19 0
sage: perm.stratum_component() == Stratum([8,6,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.three_ones_odds(odd_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in an Abelian stratum with odd order zeros and three zeros of order 1.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• odd_zeros - an even length list of odd positive integers three of which are equal to one.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = three_ones_odds([3,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10
2 10 6 5 1 8 4 7 3 9 0
sage: perm.stratum_component() == Stratum([3,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = three_ones_odds([5,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 12 9 8 1 7 3 6 10 5 4 11 0
sage: perm.stratum_component() == Stratum([5,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = three_ones_odds([7,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 14 10 9 1 4 3 6 5 12 8 11 7 13 0
sage: perm.stratum_component() == Stratum([7,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = three_ones_odds([9,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 16 13 12 1 4 3 6 5 11 7 10 14 9 8 15 0
sage: perm = three_ones_odds([3,3,3,1,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 5 7 6 4 3 8 11 13 12 10 9 14 17 1 18 16 15 0
sage: perm.stratum_component() == Stratum([3,3,3,1,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.two_ones_odds(odd_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in an Abelian stratum with odd order zeros and two zeros of order 1.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• odd_zeros - an even length list of odd positive integers two of which are equal to one.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = two_ones_odds([3,3,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12
2 5 7 6 4 3 8 11 1 12 10 9 0
sage: perm.stratum_component() == Stratum([3,3,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True
sage: perm = two_ones_odds([5,5,3,3,1,1])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
2 4 7 3 9 8 6 5 10 12 15 11 17 16 14 13 18 24 22 21 23 20 1 19 0
sage: perm.stratum_component() == Stratum([5,5,3,3,1,1], k=1).unique_component()
True
sage: _cylinder_check(perm)
True

surface_dynamics.flat_surfaces.single_cylinder.two_twos_even(real_zeros)

Returns a single cylinder permutation representative.

Returns a permutation representative of a square-tiled surface having a single vertical cylinder and a single horizontal cylinder in the even component of an Abelian stratum having two zeros of order two.

Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].

INPUT:

• real_zeros - a list of even positive integers two of which are equal to two.

EXAMPLES:

sage: from surface_dynamics import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import *
sage: from surface_dynamics.flat_surfaces.single_cylinder import _cylinder_check

sage: perm = two_twos_even([4,2,2])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11
2 8 5 3 1 10 9 6 4 11 7 0
sage: perm.stratum_component() == Stratum([4,2,2], k=1).even_component()
True
sage: _cylinder_check(perm)
True
sage: perm = two_twos_even([6,2,2])
sage: perm
0 1 2 3 4 5 6 7 8 9 10 11 12 13
2 7 6 5 3 8 4 9 11 13 10 1 12 0
sage: perm.stratum_component() == Stratum([6,2,2], k=1).even_component()
True
sage: _cylinder_check(perm)
True