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  "body": "\n## Definition\n\nLet $K_0$ and $K_1$ be oriented tame [[knots|QYEAZTYGPT]] in $S^3$.\nThey are *concordant* if there is a smoothly and properly embedded annulus $A \\cong S^1 \\times [0, 1]$ in $S^3 \\times [0, 1]$ such that $\\partial A = (K_0 \\times \\set{0}) \\cup (K_1 \\times \\set{1})$.\nThe orientation of $A$ must induce the given orientation on $K_0$ and the opposite one on $K_1$.\nA *concordance class* is an equivalence class under this relation.\n\n## Remarks\n\nIsotopic knots are concordant, since the trace of an isotopy is a concordance.\nA knot is concordant to the [[unknot|1TQ06R4Q66]] if and only if it is [[slice|P68G28E4AD]].\nThe concordance classes form an abelian group $\\mathcal{C}$ under [[knot sum|AGZSZW7GYE]], with the class of the unknot as zero.\nThe inverse of $K$ is $K^*$, the mirror image of $K$ with reversed orientation.\nExample: $K \\# K^*$ is slice, so it is concordant to the unknot.\nNon-example: the [[trefoil knot|VYTYSVHN4A]] is not concordant to the unknot, since its signature is nonzero.\nSignature is additive under knot sum, so the trefoil has infinite order in $\\mathcal{C}$.\n",
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