In this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve.
Enrique Artal Bartolo. Jorge Carmona Ruber. José Ignacio Cogolludo Agustín. "Braid monodromy and topology of plane curves." Duke Math. J. 118 (2) 261 - 278, 1 June 2003. https://doi.org/10.1215/S0012-7094-03-11823-2