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  "body": "\n## Definition\n\nLet $S^1$ be the [[circle|060W5CBZP6]] and let $\\R^3$ be [[Euclidean space|0XPR1E85W0]] of dimension $3$.\n\n1. A *knot* is a subset $K \\subset \\R^3$ that is the image of a continuous injective map $f \\colon S^1 \\to \\R^3$.\n2. Two knots $K$ and $K'$ are *equivalent* if there is a continuous family of [[homeomorphisms|0QT00PB5G2]] $h_t \\colon \\R^3 \\to \\R^3$, $t \\in [0, 1]$, with $h_0 = \\id$ and $h_1(K) = K'$. Such a family is an *ambient isotopy*.\n3. A knot is *tame* if it is equivalent to a knot that is a closed polygon with finitely many edges.\n\n## Remarks\n\nEquivalence of knots is an [[equivalence relation|0G8VEZ3WTK]], and knot theory studies the equivalence classes.\nSince $S^1$ is compact, the map $f$ is a homeomorphism onto $K$.\nMost texts consider only tame knots.\nEvery knot that is the image of a smooth embedding is tame, and every tame knot is equivalent to such a knot.\nSome texts use the $3$-sphere $S^3$ in place of $\\R^3$; this gives the same theory.\nOften a knot is also oriented, by choosing a direction along $S^1$.\n\nExample: the circle $\\set{(x, y, 0) : x^2 + y^2 = 1}$ is a knot, the image of $t \\mapsto (\\cos t, \\sin t, 0)$, which is injective on $[0, 2\\pi)$; it is a tame knot, the [[unknot|1TQ06R4Q66]].\nNon-example: the figure-eight curve $t \\mapsto (\\sin 2t, \\sin t, 0)$ is not a knot, because it is not injective: $t = 0$ and $t = \\pi$ both map to the origin.\nA [[wild knot|QYDQ28X8MA]] satisfies the first clause but not the third.\n",
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