Jones Polynomial - Definition By The Bracket

Definition By The Bracket

Suppose we have an oriented link, given as a knot diagram. We will define the Jones polynomial, using Kauffman's bracket polynomial, which we denote by . Note that here the bracket polynomial is a Laurent polynomial in the variable with integer coefficients.

First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)

,

where denotes the writhe of in its given diagram. The writhe of a diagram is the number of positive crossings ( in the figure below) minus the number of negative crossings . The writhe is not a knot invariant.

is a knot invariant since it is invariant under changes of the diagram of by the three Reidemeister moves. Invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by multiplication by under a type I Reidemeister move. The definition of the polynomial given above is designed to nullify this change, since the writhe changes appropriately by +1 or -1 under type I moves.

Now make the substitution in to get the Jones polynomial . This results in a Laurent polynomial with integer coefficients in the variable .