 
Summary: EFFECTIVE INVARIANTS OF BRAID MONODROMY AND
TOPOLOGY OF PLANE CURVES
ENRIQUE ARTAL BARTOLO, JORGE CARMONA RUBER, AND JOS´E IGNACIO
COGOLLUDO AGUST´IN
Abstract. In this paper we construct effective invariants for braid monodromy
of affine curves. We also prove that, for some curves, braid monodromy deter
mines their topology. We apply this result to find a pair of curves with conjugate
equations in a number field but which do not admit any orientationpreserving
homeomorphism.
Let C C2
be an algebraic affine curve. We say a property P(C) is an invariant
of C if it is a topological invariant of the pair (C2
, C), in other words, if P(C) =
P(C ) whenever (C2
, C) and (C2
, C ) are homeomorphic as pairs. Analogously, we
define the concept of invariants of projective algebraic curves.
Our purpose in this paper is the construction of new and effective invariants
for algebraic curves that reveal that the position of singularities is not enough to
determine the topological type of the pair (P2
