# Amphicheiral knot polynomials KT-1-H1-to-H3

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

H1: For every positive odd integer \(\ell\), there exists \(f_\ell\in\mathbb Z[z]\) such that
\[
f_\ell(s-s^{-1})f_\ell(s^{-1}-s)
\equiv(-1)^{(\ell-1)/2}\frac{s^\ell-s^{-\ell}}{s-s^{-1}}\pmod4.
\]
The quotient on the right is the Laurent polynomial
\(s^{\ell-1}+s^{\ell-3}+\cdots+s^{1-\ell}\); the congruence is coefficientwise in \((\mathbb Z/4\mathbb Z)[s^{\pm1}]\).

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.