## For (A8+3A2)+A4+A1 (nt83)
Read("GAP.txt");
beta  := [s1^-1, s2^-4, s3^-1, s2^-2, s1^-1];;
gamma := [tau^-2, s2^-1*s1, \1, \1, s3^-1*s2];;
braid := [s1^2*s3, s2^9, s3^2, s2^3, s1];;
K := 1;;
kappa := a1*a2;;
gamma_im := s2*s1^-1*s3^2*s2;      # manually computed
# rel_K := [a1*a2*(a3*a4)^2, a4*(a3*a4)^2/((a3*a4)^2*a3)];;  # manually
g_im := \1;
init_group(3);
g_im := gamma_im*Paths[1]^-1;
IsBraidID((s1^3)^g_im/dPlane, G);  # just to check
rPlane := [Braid(g_im, G, BraidRelations(s1^3, G)[2])];
create_group(: Simplified);
rels;
## Perturbations:
#: The $A_1$ point
braid := [s1^2*s3, s2^9, s3, s2^3, s1];;
init_group(0: Simplified);
rels;
qsize(g.1/g.2);
#> 6  ## This is the commutant!
simplify(g.1/g.2);
#> #I  there are 2 generators and 0 relators of total length 0
#: The $A_8$ point, perturbation of torus type
braid := [s1^2*s3, s2^3, s3^2, s2^3, s1];;
init_group(0: Simplified);
rels;
qsize(g.1/g.2);
#> 6  ## This is the commutant!
simplify(g.1/g.2);
#> #I  there are 2 generators and 0 relators of total length 0
#: The $A_8$ point, perturbation not of torus type
braid := [s1^2*s3, s2, s3^2, s2^3, s1];;
Size(init_group(3));
#> 6
#: The real $A_2$ point, perturbation not of torus type
braid := [s1^2*s3, s2^9, s3^2, s2, s1];;
Size(init_group(3));
#> 6
#: An imaginary $A_2$ point, perturbation not of torus type
#: We change the relation $\rho_\infty=\id$
braid := [s1^2*s3, s2^9, s3^2, s2^3, s1];;
init_group(3);
g_im := gamma_im*Paths[1]^-1;
IsBraidID((s1^3)^g_im/dPlane, G);  # just to check
rPlane := [Braid(g_im, G, BraidRelations(s1, G)[2])];
Size(create_group(: Simplified));
#> 6
