Bracket polynomial

From Wikipedia, the free encyclopedia

In the mathematical theory of knots, the bracket polynomial (also known as the Kauffman bracket) is used in the construction of the Jones polynomial, as well as other knot polynomials. The bracket polynomial was discovered by Louis Kauffman in 1987.

Unlike other knot polynomials the bracket polynomial is not an invariant of knots or links, but only of diagrams. Specifically, it is changed under type I Reidemeister moves. It is, however, invariant under type II and III moves. The bracket polynomial of an unoriented link diagram D is represented as <D>.

[edit] Definition

Start with any undirected link diagram L(k), and take any crossing. Label this crossing with an A and a B part (from the undercrossing, the A is on the left hand side before the crossing, with the B on the other side; and the A is on the right hand side after coming out of the crossing, with the B on the left hand side). We can define the bracket polynomial as

$\langle L(k) \rangle = A\langle L(k\ \mathrm{with\ the\ two}\ A\ \mathrm{parts\ joined\ together}) \rangle + B\langle L(k\ \mathrm{with\ the\ two}\ B\ \mathrm{parts\ joined\ together}) \rangle.\,$

$\langle \mathrm{unknot}\rangle = 1.\,$

The final bracket polynomial is the sum of A^a·B^b·d^r − 1, where a is the number of A regions, b is the number of B regions, and r is the number of unknots.