Jones polynomial

Remarks

Existence is the main theorem; uniqueness holds because changing crossings in any diagram reaches an unlink, and the relation computes the rest. Example: a one-crossing diagram of the unknot gives $L_+ = L_- = O$ and $L_0$ the two-component unlink $O \sqcup O$. The relation gives $t^{-1} - t = (t^{1/2} - t^{-1/2}) V(O \sqcup O)$, so $V(O \sqcup O) = -(t^{1/2} + t^{-1/2})$. For the trefoils, $V = t + t^3 - t^4$ for one and $V = t^{-1} + t^{-3} - t^{-4}$ for its mirror image. Non-example: the map $L \mapsto 1$ satisfies clause 1 but not clause 2, since $t^{-1} - t \ne t^{1/2} - t^{-1/2}$. In terms of the Kauffman bracket, $V_L(t) = (-A^3)^{-w(D)} \langle D \rangle$ at $A = t^{-1/4}$, where $w(D)$ is the writhe of a diagram $D$ of $L$. Mirror image changes $V_L(t)$ to $V_L(t^{-1})$.