These Thurston maps are NET maps for every choice of translation term.
They are primitive and have degree 18.
PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS
{0} {lambda1} {lambda2} {lambda1+lambda2}
These pure modular group Hurwitz classes each contain only
finitely many Thurston equivalence classes.
However, this modular group Hurwitz class contains
infinitely many Thurston equivalence classes.
The number of pure modular group Hurwitz classes
in this modular group Hurwitz class is 24.
ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM
0/6, 0/18, 1/18, 1/9, 1/3, 1/2, 2/3, 1/1, 2/2, 3/2, 2/1, 4/2, 3/1, 5/1
EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION
(-infinity,0.000000)
( 0.000000,0.492857)
( 0.506757,infinity)
The half-space computation does not determine rationality.
EXCLUDED INTERVALS FOR JUST THE SUPPLEMENTAL HALF-SPACE COMPUTATION
INTERVAL COMPUTED FOR HST OR EXTENDED HST
(-0.013818,0.012443) 0/1 EXTENDED HST
( 0.485714,0.514287) 1/2 EXTENDED HST
The supplemental half-space computation shows that these NET maps are rational.
SLOPE FUNCTION INFORMATION
NUMBER OF FIXED POINTS: 2 EQUATOR?
FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2
-3/1 1 9 No No No No
0/1 1 18 Yes Yes No No
NUMBER OF EQUATORS: 1 1 0 0
There are no more slope function fixed points.
Number of excluded intervals computed by the fixed point finder: 2158
No nontrivial cycles were found.
The slope function maps some slope to the nonslope.
The slope function orbit of every slope p/q with |p| <= 50 and
|q| <= 50 ends in either one of the above cycles or the nonslope.
If the slope function maps slope p/q to slope p'/q', then |q'| <= |q|
for every slope p/q with |p| <= 50 and |q| <= 50.
FUNDAMENTAL GROUP WREATH RECURSIONS
When the translation term of the affine map is 0:
NewSphereMachine(
"a=<1,b,b,b,b,b,b,b,b,b,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)",
"b=(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)",
"c=<1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c*d>(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)",
"d=<1,1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c>(1,2)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)",
"a*b*c*d");
When the translation term of the affine map is lambda1:
NewSphereMachine(
"a=(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)",
"b=**(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)",
"c=(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)",
"d=<1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c*d>(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)",
"a*b*c*d");
When the translation term of the affine map is lambda2:
NewSphereMachine(
"a=(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)",
"b=<1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c*d>(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)",
"c=(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)",
"d=****(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)",
"a*b*c*d");
When the translation term of the affine map is lambda1+lambda2:
NewSphereMachine(
"a=(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)",
"b=(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)",
"c=****(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)",
"d=****(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)",
"a*b*c*d");
**