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  "body": "\n## Definition\n\nLet $A = S^1 \\times [0, 1]$ be the annulus, with $S^1 = \\R / 2\\pi\\Z$ and the orientation of the $(\\theta, t)$-plane.\nThe *twist map* $T \\colon A \\to A$ is $T(\\theta, t) = (\\theta + 2\\pi t, t)$; it is the identity on both boundary circles.\n\nLet $S$ be an oriented [[surface|VGJJVBZ1MQ]] and let $c$ be a simple closed curve in the interior of $S$.\nChoose an orientation-preserving embedding $\\varphi \\colon A \\to S$ with $\\varphi(S^1 \\times \\set{1/2}) = c$.\nThe *Dehn twist* about $c$ is the [[homeomorphism|0QT00PB5G2]] $T_c \\colon S \\to S$ given by\n$$T_c(x) = \\begin{cases} \\varphi \\circ T \\circ \\varphi^{-1}(x) \u0026 \\text{if } x \\in \\varphi(A), \\\\ x \u0026 \\text{otherwise.} \\end{cases}$$\n\n## Remarks\n\nThe isotopy class of $T_c$ does not depend on the choice of $\\varphi$.\nIt depends only on the isotopy class of $c$, and it is an element of the [[mapping class group|JV5NYG913V]] of $S$.\nIts definition does not depend on an orientation of $c$, but it does depend on the orientation of $S$: reversing it replaces $T_c$ by its inverse.\n\nExample: on the torus $T^2 = \\R^2 / \\Z^2$, let $c$ be the image of the horizontal line.\nThe Dehn twist $T_c$ acts on $H_1(T^2; \\Z) = \\Z^2$ by the matrix $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$, up to the sign convention.\nSince this matrix has infinite order, $T_c$ has infinite order in the mapping class group.\n\nNon-example: if $c$ bounds a disc in $S$, then $T_c$ is isotopic to the identity, so it is trivial in the mapping class group.\nAnd on a non-orientable surface, a curve whose neighbourhood is a Möbius strip has no annulus neighbourhood, so no Dehn twist about it is defined.\n\n## Explanations\n\n### L1 human\n\nCut the surface along the curve, rotate one side through a full turn, and glue it back.\nTo make this continuous, the rotation is spread over a thin band, from no rotation on one edge to a full turn on the other.\nEvery curve that crosses the band picks up one extra lap around $c$.\n",
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