  Given a closed subvariety of an algebraic torus, the associated tropical
variety is a polyhedral fan in the space of 1-parameter subgroups of the torus
which describes the behaviour of the subvariety at infinity. We show that the
link of the origin has only top rational homology if a genericity condition is
satisfied. Our result is obtained using work of Tevelev and Deligne's theory of
mixed Hodge structures.
