= Signed Jones evaluation at minus one
{title2=$V_K(-1)=\Delta_K(-1)$}
Use the <Conway-normalized Alexander polynomial>. Choosing $t^{1/2}=i$ in the <Jones polynomial skein relation> gives $V_{L_+}(-1)-V_{L_-}(-1)=-2iV_{L_0}(-1)$. The quantities $(-1)^{\ell-1}\nabla_L(2i)$ obey the same relation, because smoothing changes the component count by one. Their <unlink> values agree, so crossing-change induction proves equality for every <link>. For a <knot>, the sign is one and $\nabla_K(2i)=\Delta_K(-1)$. This proves the signed identity, not only equality of <knot determinants>.
Back to article page
