  We construct infinitely many Legendrian links in the standard contact
$\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian
fillings. The construction is used to find links in $S^3$ that bound
topologically distinct pieces of algebraic curves in $B^4 \subset
\mathbb{C}^2$, is applied to find contact 3-manifolds with topologically
distinct symplectic fillings, and is generalized to higher dimensions.
