  Given a transverse link in the standard contact 3-sphere, we study the
contact manifold that arises as a branched double cover of the sphere. We give
a contact surgery description of such manifolds, which allows to determine the
Heegaard Floer contact invariants for some of them. By example of the knots of
Birman--Menasco, we show that these contact manifolds may fail to distinguish
between non-isotopic transverse knots. We also investigate the relation between
the Heegaard Floer contact invariants of the branched double covers and the
Khovanov homology, in particular, the transverse link invariant we introduce in
a related paper.
