HOMFLY polynomial

Remarks

Several conventions for the variables are in use; they differ by substitutions. Example: a one-crossing diagram of the unknot gives $a - a^{-1} = z \, P(O \sqcup O)$, so the two-component unlink has $P = (a - a^{-1}) z^{-1}$. Non-example: the map $L \mapsto 1$ satisfies clause 1 but violates clause 2, since $a - a^{-1} \ne z$. The substitution $a = t^{-1}$, $z = t^{1/2} - t^{-1/2}$ gives the Jones polynomial. The substitution $a = 1$, $z = t^{1/2} - t^{-1/2}$ gives the Conway-normalized Alexander polynomial.