The degree of the colored HOMFLY polynomial

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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 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 Ga11. We prove our conjecture for all positive knots. Our technique is a reformulation of the MOY state sum 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.