Connected sum of knots ($K\#J$)

The connected sum removes a short unknotted arc from each of two oriented knots and joins the four endpoints by an orientation-compatible pair of arcs.

The ambient three-sphere is also joined by connected sum, while the two knot arcs join inside it.

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.

A composite knot is a connected sum of two nontrivial knots.

A prime knot is a nontrivial knot that is not a composite knot.

A splitting sphere meets a knot transversely in two points. It is trivial when one of the resulting one-string tangles is boundary-parallel in its three-ball; otherwise it exhibits a nontrivial connected-sum decomposition.

A finite collection of disjoint splitting spheres of a knot encodes a connected sum of knots. Cut along every sphere and fill each spherical boundary with a three-ball containing a straight arc joining its two marked endpoints. The resulting oriented knots are the factors. The system is complete when its nontrivial factors are prime knots.

In a complete decomposing sphere system for a knot, a disk inside a prime-factor ball that meets the knotted arc once cuts its factor into a connected sum of knots. One side must be an unknot. Replacing the ball by the side carrying the prime factor therefore preserves the unordered factor list. A disk missing the arc cuts off an empty ball and gives the same conclusion. After a small transverse perturbation, such a replacement removes an innermost intersection circle with another sphere system. If a chosen innermost disk contains both marked points, another innermost disk, or its complementary disk when it is the only intersection, contains neither. Iterating produces a prime-factor ball disjoint from both systems. The replacement changes only the location of a joining arc in the intervening punctured ball; capping the other boundaries still recovers the same oriented factors.