{
  "statement": "H3: Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).\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 \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
  "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-H3-to-H6",
  "given_context": {
    "premise": "Let \\(L=K\\cup C\\subset S^3\\) be a smooth oriented two-component link with \\(C\\) unknotted and \\(\\operatorname{lk}(K,C)\\) odd. Let \\(\\pi:S^3\\to S^3\\) be the oriented double cover branched over \\(C\\), and give \\(K'=\\pi^{-1}(K)\\) and \\(C'=\\pi^{-1}(C)\\) the lifted orientations.\nThen \\(K'\\) is a knot and\n\\[\n\\Delta_{K'\\cup C'}(t,y^2)\\doteq\\Delta_{K\\cup C}(t,y)\\Delta_{K\\cup C}(t,-y).\n\\]\nThe variables \\(t,y\\) correspond to the knot and axis components, respectively. The two-variable Alexander polynomials are defined up to Laurent units; \\(\\doteq\\) means equality up to multiplication by \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\)."
  },
  "protocol": "single"
}