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  "body": "\n## Definition\n\nLet $K_1$ and $K_2$ be oriented tame [[knots|QYEAZTYGPT]] in $S^3$.\nChoose disjoint closed $3$-balls $B_1, B_2 \\subset S^3$ with $K_i \\subset B_i$.\nChoose a band $b \\cong [0,1] \\times [0,1]$ in $S^3$ that meets $B_1 \\cup B_2$ only in $\\set{0, 1} \\times [0,1]$.\nRequire $\\set{i-1} \\times [0,1] \\subset K_i$ for $i = 1, 2$.\nChoose the band so that the orientations of $K_1$ and $K_2$ agree with an orientation of the knot below.\nThe *knot sum* $K_1 \\# K_2$ is the oriented knot\n$$\\big( (K_1 \\cup K_2) \\setminus (\\set{0,1} \\times (0,1)) \\big) \\cup \\big( [0,1] \\times \\set{0, 1} \\big).$$\n\n## Remarks\n\nThe isotopy class of $K_1 \\# K_2$ does not depend on the balls or the band.\nExample: if $K_2$ is the [[unknot|1TQ06R4Q66]], the band slides into a trivial arc and $K_1 \\# K_2$ is isotopic to $K_1$.\nNon-example: if the band is allowed to enter $B_1$ away from $K_1$, it may wind around $K_1$.\nGluing along such a band can give a knot different from $K_1 \\# K_2$, so the condition on the band matters.\nThe sum is commutative and associative on isotopy classes of oriented knots.\nThe genus of a [[Seifert surface|K866ND3WF7]] is additive, $g(K_1 \\# K_2) = g(K_1) + g(K_2)$, so a nontrivial knot has no inverse.\nIn a diagram, the sum is drawn by placing the two diagrams side by side and replacing two facing arcs by two parallel arcs.\n",
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