Homepage of the VKCURVE package

VKCURVE is a joint project of

which we undertook in order to do the computations described in our paper "explicit presentations for exceptional braid groups" (see the file contrib/readme in the package for details).

VKCURVE is a computer algebra package which computes a presentation of the fundamental group of the complement in the complex plane of an algebraic curve, following Van Kampen's method. It is written in GAP3. If, when using this package, you find any bugs, misdesigns, problems etc... (or if you just make an interesting use of the package), please send Email to us (David Bessis or Jean Michel).

Here is brief example of the package in action:

   gap> FundamentalGroup(x^2-y^3);
   #I  there are 2 generators and 1 relator of total length 6
   1: bab=aba

   gap> FundamentalGroup((x+y)*(x-y)*(x+2*y));
   #I  there are 3 generators and 2 relators of total length 12
   1: cab=abc
   2: bca=abc

For more information you look at the manual online. You can also try VKCURVE online, an (experimental) web interface where you can directly enter a curve and get a result.

How to install VKCURVE onto your computer

To help people who are interested in VKCURVE or other GAP3 packages not ported to GAP4 I have prepared an easy-to install minimal GAP3 distribution. I recommend to use this rather than trying to install VKCURVE separately, especially since VKCURVE depends for some functionalities on some other packages being up to date.

History of the package

  • First release 0.99 on 15-2-2002
  • Version 1.0 on 1-3-2002. Now curves with Gaussian rational coefficients are accepted:
       gap> FundamentalGroup(x^2-E(4)*y^3);
       #I  there are 2 generators and 1 relator of total length 6
       1: bab=aba
  • Version 1.2 on 3-2009. Some bug fixes.