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  "body": "\n## Definition\n\nLet $K \\subset S^3$ be an oriented tame [[knot|QYEAZTYGPT]], and let $F$ be an oriented [[Seifert surface|K866ND3WF7]] of $K$ of genus $g$.\nLet $a_1, \\ldots, a_{2g}$ be oriented simple closed curves on $F$ whose classes form a basis of $H_1(F; \\Z)$.\nFor a curve $a$ on $F$, let $a^+$ be the curve $a$ pushed off $F$ slightly in the positive normal direction.\nThe *Seifert matrix* is the $2g \\times 2g$ integer matrix $V$ with entries $V_{ij} = \\operatorname{lk}(a_i, a_j^+)$, where $\\operatorname{lk}$ is the linking number.\nThe *Alexander polynomial* of $K$ is\n$$\\Delta_K(t) = \\det\\big(V - t V^{T}\\big) \\in \\Z[t, t^{-1}],$$\ntaken up to multiplication by a unit $\\pm t^{k}$ with $k \\in \\Z$.\n\n## Remarks\n\nDifferent Seifert surfaces and bases change $\\Delta_K$ only by such a unit.\nThe usual normalization, due to Conway, is the representative with $\\Delta_K(t^{-1}) = \\Delta_K(t)$ and $\\Delta_K(1) = 1$.\nExample: the [[trefoil knot|VYTYSVHN4A]] has a genus-one Seifert surface with $V = \\begin{pmatrix} -1 \u0026 1 \\\\ 0 \u0026 -1 \\end{pmatrix}$.\nThen $\\det(V - tV^T) = (t - 1)^2 + t = t^2 - t + 1$, normalized to $t - 1 + t^{-1}$.\nThe [[unknot|1TQ06R4Q66]] bounds a disk with $g = 0$, so its polynomial is the empty determinant $1$.\nNon-example of detection: the [[Conway knot|ZSAMVFPWN6]] is knotted but also has $\\Delta = 1$, so $\\Delta_K = 1$ does not imply that $K$ is the unknot.\nThe [[figure-eight knot|C01XBGAHC6]] has $\\Delta = -t + 3 - t^{-1}$.\nThe degree span of $\\Delta_K$ is at most $2g(K)$.\n",
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