- 25817606346086956476758470208850883766420383787607030339022713274189365873405154695767797745935894060883806800961601773820707673481632102376182116930107064769426021006200320503617211596800*t
+ 34918206405098505823938790072675572231488655998026261577866691497637535623661745400444768148106948876570405151317417742685700849593772165309178580940805695949542187927076864000)*Dt)
quadric_slice_pol = (
4980990673427087034113103774848375913397675011396681161337606780457883155824640000000000*t**12
- 16313074573215242896867677175985719375664055250377801991087546344967331905536000000000*t**9
- 14852779293587242300314544658084523021409425155052443959294262319432698489552764928000000*t**8
+ 18694126910889886952945780127491545129704079293214429569400282861674612412907520000*t**6
+ 32429224374768702788524801575483580065598417846595577296275963028007688596147404800000*t**5
+ 14763130935033327878568955564665179022508855828282305094488782847988800598441515915673600*t**4
- 7447056930374930458107131157447569387299331973073657492405996702806537404416000*t**3
- 18581243794708202636835504417848386599346688512251081679746508518773002589362454528*t**2
- 16116744082275656666424675660780874575937043631040306492377025123023286892432343685120*t
- 4891341219838850087826096307272910719484535278552470341569283855964428449539674077056375)
quadric_slice_crit = AA.polynomial_root(quadric_slice_pol, RIF(-0.999,-0.998))
aa = AA.polynomial_root(AA.common_polynomial(t**2 - t - 6256320), RIF(-RR(2500.7637305969961), -RR(2500.7637305969956)))
K, a = NumberField(t**2 - t - 6256320, 'a', embedding=aa).objgen()
DiffOps_x, x, Dx = DifferentialOperators(K, 'x')
iint_quadratic_alg = IVP(
dop = (
(8680468749131953125000000000000000000000*x**13
+ (34722222218750000000000000000000*a
- 8680555572048611109375000000000000000000)*x**12
- 43419899820094632213834375000000000000000*x**11
+ (
-173681336093739466250000000000000*a
+ 43420334110275534609369733125000000000000)*x**10
+ 86874920665761352031076792873375000000000*x**9
+ (347503157694622354347850650000000*a
- 86875789597407167434273839673925325000000)*x**8
- 86910050568035794059326480970966757245000*x**7
def arnoux_yoccoz(genus):
r"""
Construct the Arnoux-Yoccoz surface of genus 3 or greater.
This presentation of the surface follows Section 2.3 of
Joshua P. Bowman's paper "The Complete Family of Arnoux-Yoccoz
Surfaces."
EXAMPLES::
sage: from flatsurf import *
sage: s = translation_surfaces.arnoux_yoccoz(4)
sage: TestSuite(s).run()
sage: s.is_delaunay_decomposed()
True
sage: s = s.canonicalize()
sage: field=s.base_ring()
sage: a = field.gen()
sage: from sage.matrix.constructor import Matrix
sage: m = Matrix([[a,0],[0,~a]])
sage: ss = m*s
sage: ss = ss.canonicalize()
sage: s.cmp_translation_surface(ss)==0
True
The Arnoux-Yoccoz pseudo-Anosov are known to have (minimal) invariant
foliations with SAF=0::
sage: S3 = translation_surfaces.arnoux_yoccoz(3)
sage: Jxx, Jyy, Jxy = S3.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0]
[ 2 -2 0]
[ 0 0 2]
sage: S4 = translation_surfaces.arnoux_yoccoz(4)
sage: Jxx, Jyy, Jxy = S4.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0 0]
[ 2 -2 0 0]
[ 0 0 2 2]
[ 0 0 2 0]
"""
g=ZZ(genus)
assert g>=3
from sage.rings.polynomial.polynomial_ring import polygen
x = polygen(AA)
p=sum([x**i for i in xrange(1,g+1)])-1
cp = AA.common_polynomial(p)
alpha_AA = AA.polynomial_root(cp, RIF(1/2, 1))
field=NumberField(alpha_AA.minpoly(),'alpha',embedding=alpha_AA)
a=field.gen()
from sage.modules.free_module import VectorSpace
V=VectorSpace(field,2)
p=[None for i in xrange(g+1)]
q=[None for i in xrange(g+1)]
p[0]=V(( (1-a**g)/2, a**2/(1-a) ))
q[0]=V(( -a**g/2, a ))
p[1]=V(( -(a**(g-1)+a**g)/2, (a-a**2+a**3)/(1-a) ))
p[g]=V(( 1+(a-a**g)/2, (3*a-1-a**2)/(1-a) ))
for i in xrange(2,g):
p[i]=V(( (a-a**i)/(1-a) , a/(1-a) ))
for i in xrange(1,g+1):
q[i]=V(( (2*a-a**i-a**(i+1))/(2*(1-a)), (a-a**(g-i+2))/(1-a) ))
from flatsurf.geometry.polygon import Polygons
P=Polygons(field)
s = Surface_list(field)
T = [None] * (2*g+1)
Tp = [None] * (2*g+1)
from sage.matrix.constructor import Matrix
m=Matrix([[1,0],[0,-1]])
for i in xrange(1,g+1):
# T_i is (P_0,Q_i,Q_{i-1})
T[i]=s.add_polygon(P(edges=[ q[i]-p[0], q[i-1]-q[i], p[0]-q[i-1] ]))
# T_{g+i} is (P_i,Q_{i-1},Q_{i})
T[g+i]=s.add_polygon(P(edges=[ q[i-1]-p[i], q[i]-q[i-1], p[i]-q[i] ]))
# T'_i is (P'_0,Q'_{i-1},Q'_i)
Tp[i]=s.add_polygon(m*s.polygon(T[i]))
# T'_{g+i} is (P'_i,Q'_i, Q'_{i-1})
Tp[g+i]=s.add_polygon(m*s.polygon(T[g+i]))
for i in xrange(1,g):
s.change_edge_gluing(T[i],0,T[i+1],2)
s.change_edge_gluing(Tp[i],2,Tp[i+1],0)
for i in xrange(1,g+1):
s.change_edge_gluing(T[i],1,T[g+i],1)
s.change_edge_gluing(Tp[i],1,Tp[g+i],1)
#P 0 Q 0 is paired with P' 0 Q' 0, ...
s.change_edge_gluing(T[1],2,Tp[g],2)
s.change_edge_gluing(Tp[1],0,T[g],0)
# P1Q1 is paired with P'_g Q_{g-1}
s.change_edge_gluing(T[g+1],2,Tp[2*g],2)
s.change_edge_gluing(Tp[g+1],0,T[2*g],0)
# P1Q0 is paired with P_{g-1} Q_{g-1}
s.change_edge_gluing(T[g+1],0,T[2*g-1],2)
s.change_edge_gluing(Tp[g+1],2,Tp[2*g-1],0)
# PgQg is paired with Q1P2
s.change_edge_gluing(T[2*g],2,T[g+2],0)
s.change_edge_gluing(Tp[2*g],0,Tp[g+2],2)
for i in xrange(2,g-1):
# PiQi is paired with Q'_i P'_{i+1}
s.change_edge_gluing(T[g+i],2,Tp[g+i+1],2)
s.change_edge_gluing(Tp[g+i],0,T[g+i+1],0)
s.set_immutable()
return TranslationSurface(s)
* t**6 +
32429224374768702788524801575483580065598417846595577296275963028007688596147404800000
* t**5 +
14763130935033327878568955564665179022508855828282305094488782847988800598441515915673600
* t**4 -
7447056930374930458107131157447569387299331973073657492405996702806537404416000
* t**3 -
18581243794708202636835504417848386599346688512251081679746508518773002589362454528
* t**2 -
16116744082275656666424675660780874575937043631040306492377025123023286892432343685120
* t -
4891341219838850087826096307272910719484535278552470341569283855964428449539674077056375
)
quadric_slice_crit = AA.polynomial_root(quadric_slice_pol, RIF(-0.999, -0.998))
aa = AA.polynomial_root(AA.common_polynomial(t**2 - t - 6256320),
RIF(-RR(2500.7637305969961), -RR(2500.7637305969956)))
K, a = NumberField(t**2 - t - 6256320, 'a', embedding=aa).objgen()
DiffOps_x, x, Dx = DifferentialOperators(K, 'x')
iint_quadratic_alg = IVP(
dop=((8680468749131953125000000000000000000000 * x**13 +
(34722222218750000000000000000000 * a -
8680555572048611109375000000000000000000) * x**12 -
43419899820094632213834375000000000000000 * x**11 +
(-173681336093739466250000000000000 * a +
43420334110275534609369733125000000000000) * x**10 +
86874920665761352031076792873375000000000 * x**9 +
(347503157694622354347850650000000 * a -
86875789597407167434273839673925325000000) * x**8 -
86910050568035794059326480970966757245000 * x**7 +
(-347643678708930265539961323497102 * a +
def arnoux_yoccoz(genus):
r"""
Construct the Arnoux-Yoccoz surface of genus 3 or greater.
This presentation of the surface follows Section 2.3 of
Joshua P. Bowman's paper "The Complete Family of Arnoux-Yoccoz
Surfaces."
EXAMPLES::
sage: from flatsurf import *
sage: s = translation_surfaces.arnoux_yoccoz(4)
sage: TestSuite(s).run()
sage: s.is_delaunay_decomposed()
True
sage: s = s.canonicalize()
sage: field=s.base_ring()
sage: a = field.gen()
sage: from sage.matrix.constructor import Matrix
sage: m = Matrix([[a,0],[0,~a]])
sage: ss = m*s
sage: ss = ss.canonicalize()
sage: s.cmp(ss) == 0
True
The Arnoux-Yoccoz pseudo-Anosov are known to have (minimal) invariant
foliations with SAF=0::
sage: S3 = translation_surfaces.arnoux_yoccoz(3)
sage: Jxx, Jyy, Jxy = S3.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0]
[ 2 -2 0]
[ 0 0 2]
sage: S4 = translation_surfaces.arnoux_yoccoz(4)
sage: Jxx, Jyy, Jxy = S4.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0 0]
[ 2 -2 0 0]
[ 0 0 2 2]
[ 0 0 2 0]
"""
g = ZZ(genus)
assert g >= 3
x = polygen(AA)
p = sum([x**i for i in range(1, g + 1)]) - 1
cp = AA.common_polynomial(p)
alpha_AA = AA.polynomial_root(cp, RIF(1 / 2, 1))
field = NumberField(alpha_AA.minpoly(), 'alpha', embedding=alpha_AA)
a = field.gen()
V = VectorSpace(field, 2)
p = [None for i in range(g + 1)]
q = [None for i in range(g + 1)]
p[0] = V(((1 - a**g) / 2, a**2 / (1 - a)))
q[0] = V((-a**g / 2, a))
p[1] = V((-(a**(g - 1) + a**g) / 2, (a - a**2 + a**3) / (1 - a)))
p[g] = V((1 + (a - a**g) / 2, (3 * a - 1 - a**2) / (1 - a)))
for i in range(2, g):
p[i] = V(((a - a**i) / (1 - a), a / (1 - a)))
for i in range(1, g + 1):
q[i] = V(((2 * a - a**i - a**(i + 1)) / (2 * (1 - a)),
(a - a**(g - i + 2)) / (1 - a)))
P = ConvexPolygons(field)
s = Surface_list(field)
T = [None] * (2 * g + 1)
Tp = [None] * (2 * g + 1)
from sage.matrix.constructor import Matrix
m = Matrix([[1, 0], [0, -1]])
for i in range(1, g + 1):
# T_i is (P_0,Q_i,Q_{i-1})
T[i] = s.add_polygon(
P(edges=[q[i] - p[0], q[i - 1] - q[i], p[0] - q[i - 1]]))
# T_{g+i} is (P_i,Q_{i-1},Q_{i})
T[g + i] = s.add_polygon(
P(edges=[q[i - 1] - p[i], q[i] - q[i - 1], p[i] - q[i]]))
# T'_i is (P'_0,Q'_{i-1},Q'_i)
Tp[i] = s.add_polygon(m * s.polygon(T[i]))
# T'_{g+i} is (P'_i,Q'_i, Q'_{i-1})
Tp[g + i] = s.add_polygon(m * s.polygon(T[g + i]))
for i in range(1, g):
s.change_edge_gluing(T[i], 0, T[i + 1], 2)
s.change_edge_gluing(Tp[i], 2, Tp[i + 1], 0)
for i in range(1, g + 1):
s.change_edge_gluing(T[i], 1, T[g + i], 1)
s.change_edge_gluing(Tp[i], 1, Tp[g + i], 1)
#P 0 Q 0 is paired with P' 0 Q' 0, ...
s.change_edge_gluing(T[1], 2, Tp[g], 2)
s.change_edge_gluing(Tp[1], 0, T[g], 0)
# P1Q1 is paired with P'_g Q_{g-1}
s.change_edge_gluing(T[g + 1], 2, Tp[2 * g], 2)
s.change_edge_gluing(Tp[g + 1], 0, T[2 * g], 0)
# P1Q0 is paired with P_{g-1} Q_{g-1}
s.change_edge_gluing(T[g + 1], 0, T[2 * g - 1], 2)
s.change_edge_gluing(Tp[g + 1], 2, Tp[2 * g - 1], 0)
# PgQg is paired with Q1P2
s.change_edge_gluing(T[2 * g], 2, T[g + 2], 0)
s.change_edge_gluing(Tp[2 * g], 0, Tp[g + 2], 2)
for i in range(2, g - 1):
# PiQi is paired with Q'_i P'_{i+1}
s.change_edge_gluing(T[g + i], 2, Tp[g + i + 1], 2)
s.change_edge_gluing(Tp[g + i], 0, T[g + i + 1], 0)
s.set_immutable()
return TranslationSurface(s)