{
  "statement": "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.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).\n\nProve H6: A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
  "starting_statement": "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.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\).",
  "target_statement": "A knot \\(K\\subset S^3\\) is called amphicheiral if an orientation-reversing homeomorphism of \\(S^3\\) maps \\(K\\) to itself; the homeomorphism may preserve or reverse the knot orientation.\nFor every amphicheiral knot \\(K\\), including the unknot, there exists \\(f\\in\\mathbb Z[z]\\) such that\n\\[\n\\nabla_K(z)\\equiv f(z)f(-z)\\pmod4.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\), and the congruence is coefficientwise.",
  "case_id": "KT-1-H2-to-H6",
  "given_context": {
    "premise": "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.\nThere exists \\(\\varphi\\in\\mathbb Z[w]\\) such that\n\\[\n\\nabla_K(z)=\\varphi(z^2)^2.\n\\]\nHere \\(\\nabla_K\\) is the Conway polynomial normalized by \\(\\nabla_{\\rm unknot}=1\\)."
  },
  "protocol": "single"
}