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  "body": "\n## Definition\n\nLet $B^4$ be the closed unit ball in $\\R^4$, with boundary the $3$-sphere $S^3$.\nA tame [[knot|QYEAZTYGPT]] $K \\subset S^3$ is *smoothly slice* if there is a smoothly embedded $2$-disk $D \\subset B^4$ with $\\partial D = D \\cap S^3 = K$.\nIt is *topologically slice* if there is such a disk $D$ that is only locally flat.\nWithout qualification, *slice* means smoothly slice.\n\n## Remarks\n\nThe disk $D$ must be properly embedded, meeting $S^3$ only in its boundary.\nExample: let $K^*$ be the mirror image of $K$ with reversed orientation.\nThen $K \\# K^*$ bounds a ribbon disk, made by spinning a half of $K$ in $B^4$, so it is slice; the square knot is an instance.\nNon-example: the [[trefoil knot|VYTYSVHN4A]] is not slice, because its signature is $\\pm 2$ and the signature of a slice knot is $0$.\nBy Fox and Milnor, if $K$ is slice then $\\Delta_K(t) = f(t) f(t^{-1})$ for some $f \\in \\Z[t, t^{-1}]$, where $\\Delta_K$ is the [[Alexander polynomial|JKA25KDPQP]].\nFreedman proved that every knot with $\\Delta_K = 1$ is topologically slice.\nSome of these knots are not smoothly slice, which shows that the two notions differ.\n",
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