# Amphicheiral knot polynomials — H4

Identify \(S^3\) with \(\{(z_1,z_2)\in\mathbb C^2:|z_1|^2+|z_2|^2=1\}\). For integers \(a\ge2\) and odd \(c\), put
\[
\rho(z_1,z_2)=(e^{2\pi ic/2^a}z_1,\overline{z_2}),\qquad C=\{z_1=0\}.
\]
Let \(K\subset S^3\) be a smooth oriented knot disjoint from \(C\), with \(\rho(K)=K\), \(\rho|_K\) preserving orientation, and \(\operatorname{lk}(K,C)\) odd.
There exists \(Q\in\mathbb Z[U,V]\) such that
\[
\Delta_{K\cup C}(t,y)\doteq Q(t+t^{-1},y+y^{-1}).
\]
The variables \(t,y\) correspond to \(K,C\), respectively, and \(\doteq\) denotes equality up to a Laurent unit \(\pm t^iy^j\), \(i,j\in\mathbb Z\).