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We formulate a conjecture about the structure of `upper lines' in theexpansion of the colored Jones polynomial of a knot in powers of (q-1). TheMelvin-Morton conjecture states that the bottom line in this expansion is equalto the inverse Alexander polynomial of the knot. We conjecture that the upperlines are rational functions whose denominators are powers of the Alexanderpolynomial. We prove this conjecture for torus knots and give experimentalevidence that it is also true for other types of knots.Comment: 21 pages, 1 figure, LaTe

Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial