# Amphicheiral knot polynomials — H3

Let \(L=K\cup C\subset S^3\) be a smooth oriented two-component link with \(C\) unknotted and \(\operatorname{lk}(K,C)\) odd. Let \(\pi:S^3\to S^3\) be the oriented double cover branched over \(C\), and give \(K'=\pi^{-1}(K)\) and \(C'=\pi^{-1}(C)\) the lifted orientations.
Then \(K'\) is a knot and
\[
\Delta_{K'\cup C'}(t,y^2)\doteq\Delta_{K\cup C}(t,y)\Delta_{K\cup C}(t,-y).
\]
The variables \(t,y\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \(\doteq\) means equality up to multiplication by \(\pm t^iy^j\), \(i,j\in\mathbb Z\).