The Jones Polynomial

Historical Background

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:

VL(t) = f[L] · t-1/4


where VL(t) is the Jones Polynomial and f[L] is the Kauffman Polynomial.

Setting Up

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.

Performing the Calculations

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.


VL-(t) = (t-1 · VL+(t) - (t1/2 - t-1/2) · VL0) / t


Then, the crossing is changed to the two other types, L+ and L0, and the polynomial invariant of the whole knot is determined for each. In the case of L+, a type II Reidemeister move followed by a type I Reidemeister move leads to the unknot. Therefore, the value of VL+(t) is 1.

VL+(t) = 1


A similar technique leads to the oriented Hopf Link in the case of L0, and some busy work leads to the formula below for VL0(t).

VL0(t) = -t - t-1


Substituting these values into the original skein relation returns the final result for the Jones Polynomial of the left-handed trefoil knot.

VL(t) = t-1 + t-3 - t-4


Conclusion

The discovery of the Jones Polynomial provoked a wave of excitement that eventually led to the discovery of the HOMFLY Polynomial.