  We investigate the local contribution of the braid monodromy factorization in
the context of the links obtained by the closure of these braids. We consider
plane curves which are arrangements of lines and conics as well as some
algebraic surfaces, where some of the former occur as local configurations in
degenerated and regenerated surfaces in the latter. In particular we focus on
degenerations which involve intersection points of multiplicity two and three.
We demonstrate when the same links arise even when the local arrangements are
different.
