In the last post we investigated the linking number and writhe. These were numerical invariants of oriented links and framed knots. Now I will introduce new invariants which take their values as polynomials.

For a given crossing, we can perform an operation called resolving or smoothing the crossing. We can do this in two ways.

0-smoothing

1-smoothing

Let us suppose that there is a polynomial invariant of links in variables so that concentrating on a neighborhood of a crossing in a diagram for , we have that the following relation, called the skein relation, holds.

Skein Relation

Performing smoothings on all crossings reduces a link diagram to some number of circles in the plane. Let’s require that adding a circle to a link diagram gives . Finally we require a normalization, that for the empty link . From this we can deduce that the bracket of circles is .

We need to check invariance under Reidemeister moves. Let’s start with Reidemeister 2.

Reidemeister 2 Calculation

Thus, in order for the bracket to be invariant we must have and . Solving for in terms of , we get .

The nice thing now is that Reidemeister 3 comes along for free by using invariance under Reidemeister 2.

Reidemeister 3 Calculation

Performing Reidemeister 1 on the other hand does not leave the bracket invariant. However, we can see that opposite Reidemeister 1 moves cancel so that the bracket is invariant under the framed Reidemeister 1 move.

Reidemeister 1 Calculation

Consequently, the bracket is an invariant of framed links whose values are polynomials in and . To calculate it, take a blackboard diagram for the framed link and apply the skein relation, the circle relation and the normalization relation until you reach the answer.

The bracket was introduced by Kauffman as an elementary way to define Jones’ polynomial, an invariant of oriented links which was originally derived using some difficult algebra. We can define the Jones’ polynomial by . Here, the total writhe is the sum of signs of all crossings in the diagram and it is this factor which makes now invariant under Reidemeister 1 moves.

The Kauffman bracket and Jones’ polynomial are very closely related, in a similar way to how the writhe and linking numbers are closely related. Following the discovery of the Jones’ polynomial, there was a great deal of interest in knot theory. The Jones’ polynomial showed new connections between topology on the one hand and representation theory and quantum physics on the other.

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