TITLE: Balanced presentation of the fundamental group of a Seifert fiber space
QUESTION [7 upvotes]: Is there any readily available reference for a balanced presentation of the fundamental group in terms of the classifying invariants of an arbitrary Seifert fibered space? I can't find it, and the usual presentation coming from the classification resists GAP (checked on four fibrations with three exceptional fibers over the sphere, got two generators and three relations each time).

REPLY [9 votes]: You can get a balanced presentation of the group from a Heegaard diagram. The paper of Boileau-Zieschang, Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math. 76 (1984), no. 3, 455–468 shows how to find such a diagram. Following their discussion (too involved to reproduce here) you get a presentation that is determined by the Seifert invariants.
If you only care about the group, then their lemma 1.5 shows how to go from the `usual' presentation (which is not balanced) to a balanced one.
Except for some small examples, I think that these are in fact minimal (with respect to the number of generators) presentations of the fundamental group.