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