  The colored HOMLFY polynomial is an important knot invariant depending on two
variables $a$ and $q$. We give bounds on the degree in both $a$ and $q$
generalizing Morton's bounds \cite{Mo86} for the ordinary HOMFLY polynomial.
Our bounds suggest that the degree detects certain incompressible surfaces in
the knot complement and perhaps more generally features of the $SL(N)$
character varieties of the knot group. We formulate a precise conjecture along
these lines generalizing the slope conjecture of Garoufalidis \cite{Ga11}. We
prove our conjecture for all positive knots.
  Our technique is a reformulation of the MOY state sum \cite{MOY98} using
$q$-analogues of Ehrhart polynomials. As a direct application we explicitly
compute the $r$ coefficients of $r$-colored HOMFLY polynomial of any positive
braid.
