The Slope Conjecture relates a quantum knot invariant, (the degreeof the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in theknot complement). The degree of the colored Jones polynomial can be computedby a suitable (almost tight) state sum and the solution of a correspondingquadratic integer programming problem. We illustrate this principlefor 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 of one sector.
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