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  "body": "\n## Definition\n\nLet $M$ be a closed orientable [[3-manifold|R20TB2KEQQ]] and let $K \\subset M$ be a smooth [[knot|QYEAZTYGPT]], that is, a smoothly embedded circle.\nLet $N(K)$ be a closed tubular neighbourhood of $K$, a solid torus, and let $E = M \\setminus \\operatorname{int} N(K)$ be the knot exterior, with boundary torus $\\partial E = \\partial N(K)$.\nLet $\\alpha \\subset \\partial E$ be an essential simple closed curve, called the *slope*.\nLet $V = D^2 \\times S^1$ and let $h \\colon \\partial V \\to \\partial E$ be a [[homeomorphism|0QT00PB5G2]] with $h(\\partial D^2 \\times \\set{1}) = \\alpha$.\nThe *Dehn surgery* on $K$ with slope $\\alpha$ is the manifold\n$$M_K(\\alpha) = E \\cup_h V.$$\nIts homeomorphism type depends only on the isotopy class of $\\alpha$ in $\\partial E$, not on its orientation.\nFor a [[link|1A5WRCNTM3]] one does this on each component, with one slope for each.\n\n## Remarks\n\nFor a knot $K$ in $S^3$, let $\\mu$ be a meridian, bounding a disc in $N(K)$, and $\\lambda$ a longitude that is null-homologous in $E$, both oriented.\nEach slope is isotopic to a curve representing $p\\mu + q\\lambda$ in $H_1(\\partial E; \\Z)$ with $p, q$ coprime, and the slope is labelled by $p/q \\in \\Q \\cup \\set{\\infty}$.\nThe slope $\\infty = 1/0$ gives back $S^3$; a slope with $q = \\pm 1$ is *integral*.\n\nExample: let $U$ be the unknot in $S^3$.\nIts exterior is a solid torus in which $\\lambda$ bounds a disc.\nSurgery with slope $0$, that is $\\alpha = \\lambda$, glues two solid tori so that their meridians match, and the result is $S^2 \\times S^1$.\nSurgery on $U$ with slope $p/q$, $p \\ne 0$, gives a lens space $L(p, q)$, up to the orientation convention.\n\nNon-example: a curve $\\alpha$ that bounds a disc in $\\partial E$ is not a slope.\nNo homeomorphism $h$ can send $\\partial D^2 \\times \\set{1}$ to it, because a homeomorphism of tori sends essential curves to essential curves.\n\n## Explanations\n\n### L1 human\n\nDrill out a thin tube around a knot and fill the hole with a solid tube glued in a new way.\nThe only choice is which loop on the boundary torus becomes the boundary of a disc of the new tube.\nThis choice is a fraction, and different fractions usually give different spaces.\n",
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