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  "body": "\n## Definition\n\nLet $\\Sigma$ be an oriented integral [[homology 3-sphere|HK8NS02NNE]], let $K \\subset \\Sigma$ be a [[knot|QYEAZTYGPT]] and let $n$ be an integer.\nLet $\\Sigma_{K, 1/n}$ be the result of [[Dehn surgery|ZA405B0Z1S]] on $K$ with coefficient $1/n$; it is again an oriented integral homology $3$-sphere.\nLet $\\Delta_K(t)$ be the [[Alexander polynomial|JKA25KDPQP]] of $K$, normalised so that $\\Delta_K(t^{-1}) = \\Delta_K(t)$ and $\\Delta_K(1) = 1$.\n\nThe *Casson invariant* is the unique function $\\lambda$ from oriented integral homology $3$-spheres, up to orientation-preserving homeomorphism, to $\\Z$ such that:\n\n1. $\\lambda(S^3) = 0$, and\n2. for every $\\Sigma$, $K$ and $n$ as above,\n$$\\lambda(\\Sigma_{K, 1/(n+1)}) - \\lambda(\\Sigma_{K, 1/n}) = \\tfrac{1}{2} \\Delta_K''(1).$$\n\nCasson proved that such a function exists and is unique.\n\n## Remarks\n\nExample: for a trefoil knot $K$ in $S^3$, $\\Delta_K(t) = t - 1 + t^{-1}$, so $\\Delta_K''(t) = 2 t^{-3}$ and $\\tfrac12 \\Delta_K''(1) = 1$.\nSince $S^3_{K, 1/0} = S^3$, clause 2 with $n = 0$ gives $\\lambda(S^3_{K, 1}) = 1$.\nFor the right-handed trefoil, $S^3_{K, 1}$ is the Poincaré homology sphere with the orientation opposite to that of $\\Sigma(2, 3, 5)$, and $\\lambda(\\Sigma(2, 3, 5)) = -1$.\n\nFurther properties: $\\lambda(-\\Sigma) = -\\lambda(\\Sigma)$, $\\lambda$ is additive under connected sum, and $\\lambda(\\Sigma)$ is congruent modulo $2$ to the Rokhlin invariant of $\\Sigma$.\n",
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