  We generalize the Morton-Franks-Williams inequality to the colored
$\mathfrak{sl}(N)$ link homology defined in arXiv:0907.0695, which gives
infinitely many new bounds for the braid index and the self linking number. A
key ingredient of our proof is a composition product for the general MOY graph
polynomial, which generalizes that of Wagner arXiv:math/0702230v1.
