#% -*- GAP -*-
#% This file is marked for TeX processing
#[caption]"Computing $\pi_1$ for sextics with a type $\bD_5$ point"
#================================================================
#: The computation of the fundamental groups of plane sextics with a%
#: type $\bD_5$ singularity.
#----------------------------------------------------------------
Reread("pi1sc.txt");   #: (see \file{pi1sc})
#: Skeletons with a bibigonal insertion
lbb := [x^2,x^2*y*x^2,x^2*y*x*y*x^2,x^2*y*(x*y)^2*x^2,x^2*y*x*(y*x^2)^2];
bb := [(G.1*G.2)^2, G.2*G.3/rho^4];    #: fixed relations \eqref{eq.D5.bbg}
group := function(arg) return DS(AA(lbb, arg), bb); end;
size := function(arg) return Size(CallFuncList(group, arg)); end;
#: Genuine tetragon: use "inf(p);" and then "size2(s0,s1,s2,s3);"%
#: where $p$ is a path from $e=e\subroman2$ to $e\subroman1$%
#: and $s_i$ is the braid index of $\reg_-(e\ra(\y\x)^i\x)$,%
#: \cf. \kwd*{group*h} in \file{E6}
lbg := List([0..3], i -> (y*x)^i*x); #%
bg := [];
inf := function(p)    #: initialize relations \eqref{eq.D5.1}--\eqref{eq.D5.inf}
 bg := bm(p, [G.1*G.2*G.3/(G.3*G.1*G.2), G.1*G.2]);
 bg := [bg[1], (G.1*G.2)^2*bg[2]/rho^4, (G.1*G.2)^2*G.3/(G.3*(G.1*G.2)^2)];
end;
group2 := function(arg) return DS(AA(lbg, arg), bg); end;
size2 := function(arg) return Size(CallFuncList(group2, arg)); end;
#================================================================
#[endinput]
