= Prime decomposition of knots
= Schubert prime decomposition theorem
{c}
{synonym}
Every nontrivial oriented <knot> is a finite <connected sum of knots> of <prime knots>, uniquely up to permutation and oriented <link isotopy>. Existence follows by induction from <additivity of Seifert genus>. For uniqueness, compare complete <decomposing sphere systems for a knot> and remove their intersection <circles> by the <prime-ball exchange lemma>. Once a prime ball is disjoint from both systems, its factor can be removed from both; induction then matches all factors.
Back to article page
