{
  "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 H4: Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
  "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": "Identify \\(S^3\\) with \\(\\{(z_1,z_2)\\in\\mathbb C^2:|z_1|^2+|z_2|^2=1\\}\\). For integers \\(a\\ge2\\) and odd \\(c\\), put\n\\[\n\\rho(z_1,z_2)=(e^{2\\pi ic/2^a}z_1,\\overline{z_2}),\\qquad C=\\{z_1=0\\}.\n\\]\nLet \\(K\\subset S^3\\) be a smooth oriented knot disjoint from \\(C\\), with \\(\\rho(K)=K\\), \\(\\rho|_K\\) preserving orientation, and \\(\\operatorname{lk}(K,C)\\) odd.\nThere exists \\(Q\\in\\mathbb Z[U,V]\\) such that\n\\[\n\\Delta_{K\\cup C}(t,y)\\doteq Q(t+t^{-1},y+y^{-1}).\n\\]\nThe variables \\(t,y\\) correspond to \\(K,C\\), respectively, and \\(\\doteq\\) denotes equality up to a Laurent unit \\(\\pm t^iy^j\\), \\(i,j\\in\\mathbb Z\\).",
  "case_id": "KT-1-H3-to-H4",
  "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"
}