  The Slope Conjecture relates a quantum knot invariant, (the degree of the
colored Jones polynomial of a knot) with a classical one (boundary slopes of
incompressible surfaces in the knot complement). The degree of the colored
Jones polynomial can be computed by a suitable (almost tight) state sum and the
solution of a corresponding quadratic integer programming problem. We
illustrate this principle for a 2-parameter family of 2-fusion knots. Combined
with the results of Dunfield and the first author, this confirms the Slope
Conjecture for the 2-fusion knots.
