The Jones Polynomial was discovered by Vaughan F. R. Jones in 1984. Unlike the Alexander Polynomial, the Jones Polynomial distinguishes between a knot and its mirror image. The Jones Polynomial is essentially the same as the Kauffman Polynomial. In fact, they can be derived from each other through this equation:
The Jones Polynomial is derived from an oriented knot diagram through two basic rules:
Rule 1: VU(t) = 1
Rule 2: t-1 · VL+(t) - t · VL-(t) = (t1/2 - t-1/2) · VL0(t)
The first rule is the rule for the oriented unknot. The Jones Polynomial invariant for the unknot is the popular value 1. The second rule in calculating the Jones Polynomial is the skein relation. The terms VL+(t), VL-(t), and VL0(t) correspond to the polynomial invariants in terms of t of the knot if a particular crossing were changed as shown below.
The first step in calculating the Jones Polynomial is to select a crossing in the oriented knot diagram to begin with. The type of crossing is then determined to be L+, L-, or L0. In the example of the left-handed trefoil knot, a crossing of type L- has been selected. The skein relation is solved in terms of VL-(t), the polynomial invariant of the knot given this type of crossing.
The discovery of the Jones Polynomial provoked a wave of excitement that eventually led to the discovery of the HOMFLY Polynomial.