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  "body": "\n## Definition\n\nLet $\\mathcal{K}$ be the set of [[knots|QYEAZTYGPT]] in $\\R^3$, and let $S$ be a set.\nA *knot invariant* with values in $S$ is a map $f \\colon \\mathcal{K} \\to S$ such that $f(K) = f(K')$ whenever the knots $K$ and $K'$ are equivalent.\n\n## Remarks\n\nEquivalently, a knot invariant is a map on the set of equivalence classes of knots.\nThe values may be numbers, polynomials, groups up to isomorphism, or homology theories up to isomorphism.\nExample: the [[crossing number|SWJZ4QRF9W]] is a knot invariant with values in $\\N$, because it is a minimum taken over the diagrams of all knots equivalent to $K$.\nNon-example: the number of crossings of one chosen diagram of $K$ is not an invariant of $K$.\nIndeed, a twist in a diagram of the unknot adds a crossing without changing the knot.\nAn invariant is *complete* if $f(K) = f(K')$ implies that $K$ and $K'$ are equivalent.\nThe same notion is used for links.\n",
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