Notes for AKT-090910-1/0:45:11

The Kauffman bracket is completely determined by the relations:

1. [math]\displaystyle{ \left\langle\slashoverback\right\rangle=A\left\langle\hsmoothing\right\rangle + B \left\langle\smoothing\right\rangle }[/math]
2. [math]\displaystyle{ \left\langle \bigcirc^k \right\rangle = d^k }[/math]

These suffice because for any knot or link diagram we can apply the first relation recursively until we eliminate all the crossings and end up with a disjoint union of unknots. For instance, we get: [math]\displaystyle{ \left\langle \HopfLink \right\rangle = A^2d^2 + 2ABd + B^2d^2 }[/math]