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

The Kauffman bracket is completely determined by the relations:

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\slashoverback\right\rangle=A\left\langle\hsmoothing\right\rangle + B \left\langle\smoothing\right\rangle }
2. $\left\langle \bigcirc ^{k}\right\rangle =d^{k}$

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:

Failed to parse (unknown function "\HopfLink"): {\displaystyle \left\langle \HopfLink \right\rangle = A^2d^2 + 2ABd + B^2d^2}