Conway polynomial (knot theory, ${\nabla }_L(z)$)

The Conway polynomial is normalized by ${\nabla }_U=1$ for the unknot, zero for a split unlink with more than one component, and the skein relation ${\nabla }_{L_{+}}-{\nabla }_{L_{-}}=z{\nabla }_{L_0}$. For a knot, the symmetric Alexander polynomial with value one at $t=1$ is ${\Delta }_K(t)={\nabla }_K(t^{1/2}-t^{-1/2})$. This fixes the Laurent-unit ambiguity of the Seifert-matrix presentation of the Alexander module.

The symmetric Alexander polynomial normalized by the Conway polynomial is uniquely fixed for a knot. Its sign matters in the signed Jones evaluation at minus one, whereas the knot determinant retains only the absolute value.