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  "body": "\n## Definition\n\nLet $K$ be a tame [[knot|QYEAZTYGPT]] in $S^3$.\nThe knot $K$ is *prime* if:\n\n1. $K$ is not the [[unknot|1TQ06R4Q66]];\n2. whenever $K$ is isotopic to a [[knot sum|AGZSZW7GYE]] $K_1 \\# K_2$, one of $K_1$ and $K_2$ is the unknot.\n\n## Remarks\n\nEquivalently, $K$ is nontrivial and every $2$-sphere in $S^3$ that meets $K$ transversely in two points bounds a ball meeting $K$ in an unknotted arc.\nExample: the [[trefoil knot|VYTYSVHN4A]] has Seifert genus $1$.\nGenus is additive under knot sum and only the unknot has genus $0$, so a decomposition $K_1 \\# K_2$ forces one summand to be trivial.\nNon-example: the granny knot, the sum of two trefoils, fails clause 2.\nThe unknot fails clause 1.\nBy Schubert's theorem, every nontrivial knot is a knot sum of finitely many prime knots, unique up to order and isotopy.\n",
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