{
  "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 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": "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": "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-H1-to-H4",
  "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"
}