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#[caption]"Computing $\pi_1$ for sextics with a type $\bE_6$ point"
#================================================================
#: The computation of the fundamental groups of plane sextics with a%
#: type $\bE_6$ singularity.
#----------------------------------------------------------------
Reread("pi1sc.txt");   #: (see \file{pi1sc})
#: Relations at infinity and the slope
fixed := [rho^4/(G.1*G.2)^3];  kappa := (G.1*G.2)^3;
#: Hexagon with a loop, irreducible curves (see \autoref{arg.E6.loop})
loop := [(G.1*G.2)^-2*G.1*(G.1*G.2)^2/G.3];  #: relation from the inner loop
c1l := [x, x^2, x*y*x^2, x*(y*x^2)^2];       #: chains to be used
group1l := function(arg) return DS(AA(c1l, arg), loop); end;
size1l := function(arg) return Size(CallFuncList(group1l, arg)); end;
#: Hexagon with a loop, reducible curves (see \autoref{arg.E6.loop})
loop2 := [loop[1], kappa];                   #: add a central element
c2l := [x, x^2, x*y*x^2, (x*y)^2*x^2];       #: chains to be used
group2l := function(arg) return DS(AA(c2l, arg), loop2); end;
size2l := function(arg) return Size(CallFuncList(group2l, arg)); end;
#: Other sextics with two type $\bE$ points (see \autoref{arg.two.E})
ce := [x, y*x^2];                            #: chains to be used
group1e := function(arg) return DS(AA(ce, arg), E6(x^2*y)); end;
#: Genuine hexagon (see \autoref{arg.E6.hex})
ch := List([0..5], i -> (y*x)^i*x);          #: chains to be used
group0h := function(arg) return DS(AA(ch, arg)); end;
#: One monogonal region inside $B\sminus\rR1$ (see \autoref{arg.E6.1.monogon})
loop1h := s -> [RA((y*x^2)^3, 1), (G.2*G.3)^(s[1]/2)];
group1h := function(arg) return DS(AA(ch, arg), loop1h(arg)); end;
#: Two monogonal regions inside $B\sminus\rR1$ (see \autoref{arg.E6.2.monogon})
loop2h := s -> [RA((y*x^2)^2, 1), (G.2*G.3)^(s[1]/2)];
group2h := function(arg) return DS(AA(ch, arg), loop2h(arg)); end;
#================================================================
#[endinput]
