Knots and Tropical Curves

A sequence of rational functions in a variable q is q-holonomic if it satisfies a linear recursion with coefficients polynomials in q and qn. In the paper, we assign a tropical curve to every q-holonomic sequence, which is closely related to the degree of the sequence with respect to q. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Conjecture and the Slope Conjecture (which relate the Jones polynomial of a knot and its parallels to the SL(2, C) character variety and to slopes of incompressible surfaces). Our discussion predicts that the tropical curve is dual to a Newton subdivision of the A-polynomial of the knot. We compute explicitly the tropical curve for the 41, 52 and 61 knots and verify the above prediction.