# Amphicheiral knot polynomials KT-1-H0-to-H2

## 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 H2: Let \(K\subset S^3\) be a smooth oriented knot admitting an orientation-reversing diffeomorphism \(h:S^3\to S^3\) with \(h^2=\operatorname{id}\), \(h(K)=K\), and \(h|_K\) preserving orientation.
There exists \(\varphi\in\mathbb Z[w]\) such that
\[
\nabla_K(z)=\varphi(z^2)^2.
\]
Here \(\nabla_K\) is the Conway polynomial normalized by \(\nabla_{\rm unknot}=1\).

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.