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  "body": "\n## Statement\n\nLet $p, q \\ge 2$ be coprime integers and let $T(p, q)$ be the [[torus knot|NS60VV786W]] of type $(p, q)$.\nFor a knot $K \\subset S^3 = \\partial B^4$, consider the compact oriented surfaces smoothly and properly embedded in $B^4$ with boundary $K$.\nThe *slice genus* $g_4(K)$ is the least genus of such a surface.\nThen\n$$g_4(T(p, q)) = u(T(p, q)) = \\frac{(p-1)(q-1)}{2},$$\nwhere $u$ is the [[unknotting number|PW9D9QKAD4]].\n\n## Remarks\n\nFor any knot, $g_4(K) \\le u(K)$: each crossing change can be realized by adding a handle to a surface in $B^4$.\nAn explicit unknotting shows $u(T(p, q)) \\le (p-1)(q-1)/2$.\nThe hard part is the lower bound $g_4(T(p, q)) \\ge (p-1)(q-1)/2$.\nKronheimer and Mrowka proved it in 1993 using gauge theory.\nRasmussen later gave a combinatorial proof with his invariant from [[Khovanov homology|XB24W24B20]].\nExample: for the [[trefoil knot|VYTYSVHN4A]] $T(2, 3)$, the formula gives $g_4 = u = 1$.\nMilnor asked the question in his study of singular points of complex plane curves, where torus knots appear as links of singularities.\n\n## Explanations\n\n### L1 human\n\nA torus knot is wound on a doughnut, going $p$ times one way and $q$ times the other.\nThe theorem gives the exact number of crossing changes needed to untie it: $(p-1)(q-1)/2$.\nIt also says that no surface in the four-dimensional ball bounded by it can have fewer handles than this.\nBoth quantities are easy to bound from one side and very hard to bound from the other.\n\n### L2 human\n\nThe upper bounds are elementary, but lower bounds on the slice genus need deep tools.\nKronheimer and Mrowka obtained the bound from gauge theory, through lower bounds on the genus of embedded surfaces in $4$-manifolds.\nRasmussen's $s$-invariant, the Ozsváth–Szabó $\\tau$-invariant from [[knot Floer homology|5P608WRETM]], and the signature give bounds of the form $g_4(K) \\ge \\abs{s(K)}/2$ and $g_4(K) \\ge \\abs{\\tau(K)}$.\nFor torus knots, the $s$- and $\\tau$-invariants are large enough to give the exact value.\n",
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