# Amphicheiral knot polynomials KT-1-H0-to-H3

## Setting

You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

Prove 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 \(\pm t^iy^j\), \(i,j\in\mathbb Z\).

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.