# Amphicheiral knot polynomials KT-1-H0-to-H1

## 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 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 \((\mathbb Z/4\mathbb Z)[s^{\pm1}]\).

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.