{
  "statement": "H1: For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).\n\nProve 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\\).",
  "starting_statement": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
  "target_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\\).",
  "case_id": "KT-1-H1-to-H2",
  "given_context": {
    "premise": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\)."
  },
  "protocol": "single"
}