Existence and uniqueness of prime decomposition for oriented knots

Prove existence by induction on the positive integer $g(K)$. If $K$ is prime there is nothing to split. Otherwise the definition provides $K=J\mathrm{\#}L$ with both factors nontrivial. By [F1], $g(J),g(L)\ge 1$ and $g(J)+g(L)=g(K)$, so both genera are strictly smaller. The induction hypothesis gives each factor a finite prime decomposition; concatenate these lists using [F3]. For genus one the nonsimple case is impossible by the same inequalities. This establishes finite existence for every positive genus.

Realize any finite connected-sum presentation by its standard separating spheres: take the gluing sphere in each two-ball construction, and place later summand constructions away from earlier small gluing collars. The resulting finite disjoint sphere system cuts the knot into arcs whose capped knots are exactly the original factors by [F2]. Conversely successive leaf regions of such a system give its connected-sum presentation, again by [F2] and associativity. Thus every prime presentation gives a prime sphere system with the same oriented factors.

For two prime presentations of $K$, use step 2.1 to obtain their two prime sphere systems in the same oriented knot sphere, transporting the standard constructions through the given ambient isotopies. Apply [F2]'s marked cut-and-paste comparison. Its replacements preserve oriented capped factors and reduce intersections until a common disjoint refinement compares the two lists. Therefore the lists have the same length and the same oriented prime knot classes, up to permutation. This is the full existence and uniqueness assertion.