The Jones Polynomial is actually a special case of a more general polynomial invariant called the HOMFLY polynomial, the skein polynomial, or the LYMPH-TOFU polynomial, depending on who you ask. These three are equivalent polynomials with slight variation in notation.
The Jones polynomial can be obtained from the Kauffman Bracket as follows:
Let w(D) denote the writhe of an oriented link diagram D of a link L (that is, the sum of the signs of the crossings of D). Let <D> be the Kauffman bracket of D (a laurent polynomial in the variable A). Then, with the substitution A2 = t½, the Jones polynomial of the link is:
VL(t) = (-A)-3w(D)<D>
Some particular evaluations of the Jones polynomial of interest at roots of unity are:
- VL(1) = (-2)#L-1
- VL(e2(pi)i/3) = (-1)#L-1
- VL(e4(pi)i/3) = 1
- VL(i) = (-sqrt(2))#L-1(-1)Arf(L) (or 0 where Arf(L) is undefined)
- VL(-1) = DeltaL(-1)
Here, #L denotes the linking number of a link, Arf(L) denotes the Arf invariant, and DeltaL denotes the Conway polynomial.
It is not known whether the Jones polynomial distinguishes the unknot (that is, it is unknown whether there is a nontrivial link (knot) with Jones polynomial equal to 1). Finding such a knot, or proving that none such exist, is an important open question in knot theory.
Sources: Kawauchi, Akio. A Survey of Knot Theory. Birkhauser Verlag: 1996.
Lickorish, W.B. Raymond. An Introduction to Knot Theory. Springer: 1997.