The Bracket and Kauffman Polynomial

## Historical Background

The Bracket Polynomial is actually a part of a publication by Kauffman as a precursor for the Jones Polynomial. The Bracket Polynomial is invariant under a regular isotopy but it is not isotopy invariant because it is not invariant under a type I Reidemeister move. Kauffman, however, found a term to multiply the Bracket Polynomial by that would make the polynomial isotopy invariant and make the polynomial equivalent to the Jones Polynomial.

## Setting Up

The Bracket Polynomial is derived from a non-oriented knot diagram. The Bracket Polynomial for a knot diagram D is written as <D> and is calculated using these three rules:

<U> = 1
<DU> = (A2 + A-2) · <D>
<C> = A · <L> + A-1 · <R>

Where <U> refers to the polynomial of the unknot, <DU> refers to the trivial link of a knot D and the unknot, and <C>, <L>, and <R> refer to the polynomials of the knot if a selected crossing were changed to each of the crossings diagrammed below.

## Performing Calculations

The idea behind the Bracket Polynomial is to break down a knot into a trivial link of unknots. A crossing is selected, like the one below.

The knot above is broken down as shown here.

 L R

<D> = A · <L> + A-1 · <R>

Now, L can be broken down further.

 LL LR

<D> = A · (A · <LL> + A-1 · <LR>) + A-1 · <R>

After an exhaustive expansion, one would arrive at the following Bracket Polynomial for the left-handed trefoil:

<K> = A7 - A3 - A-5

## Conclusion

Due to the fact that the Bracket Polynomial is invariant under only a regular isotopy and not under a type I Reidemeister move, a knot cannot be simplified while its polynomial is being calculated. This leads to an exhaustive expansion and tedious simplification, but it can be made an invariant under a type I Reidemeister move in this equation:

f[L] = (-A)-3 · w(D) · <D>

In this equation, w(D) is the writhe of the knot, and f[L] is the Kauffman Polynomial.