  We describe a normal surface algorithm that decides whether a knot, with
known degree of the colored Jones polynomial, satisfies the Strong Slope
Conjecture. We also discuss possible simplifications of our algorithm and state
related open questions. We establish a relation between the Jones period of a
knot and the number of sheets of the surfaces that satisfy the Strong Slope
Conjecture (Jones surfaces). We also present numerical and experimental
evidence supporting a stronger such relation which we state as an open
question.
