On the Jones Polynomial in the Solid Torus

We introduce an infinite class of elementary knots in the solid torus, together with general recursive and explicit formulas of the values of these knots under the generalized invariant of Jones polynomial to the solid torus. These values can be used as an infinite set of initial data for this invariant. We also introduce a procedure of resolving certain knots called spiral knots into these elementary knots. We show that our explicit formulas involve exactly n + 1 terms for an elementary knot with n crossings, which reduces the calculations needed to compute the invariant for spiral knots and arbitrary knots and links in the solid torus.