{
  "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 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": "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": "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-H1-to-H6",
  "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"
}