  The conormal lift of a link $K$ in $\R^3$ is a Legendrian submanifold
$\Lambda_K$ in the unit cotangent bundle $U^* \R^3$ of $\R^3$ with contact
structure equal to the kernel of the Liouville form. Knot contact homology, a
topological link invariant of $K$, is defined as the Legendrian homology of
$\Lambda_K$, the homology of a differential graded algebra generated by Reeb
chords whose differential counts holomorphic disks in the symplectization $\R
\times U^*\R^3$ with Lagrangian boundary condition $\R \times \Lambda_K$.
  We perform an explicit and complete computation of the Legendrian homology of
$\Lambda_K$ for arbitrary links $K$ in terms of a braid presentation of $K$,
confirming a conjecture that this invariant agrees with a previously-defined
combinatorial version of knot contact homology. The computation uses a double
degeneration: the braid degenerates toward a multiple cover of the unknot which
in turn degenerates to a point. Under the first degeneration, holomorphic disks
converge to gradient flow trees with quantum corrections. The combined
degenerations give rise to a new generalization of flow trees called multiscale
flow trees. The theory of multiscale flow trees is the key tool in our
computation and is already proving to be useful for other computations as well.
