Compound knot

In the mathematical field of knot theory , a compound knot is a knot that can be decomposed (in the manner described below) as the connected sum of two nontrivial knots . Nodes that are not assembled are called prime nodes . Each node can be decomposed as a connected sum of prime nodes.

Connected sum of nodes

The formation of the connected sum (also: node sum or connected sum ) of two nodes is described by the following sequence of diagrams:

Disjoint plane projections of two nontrivial nodes.

Find a rectangle in the plane with two opposite edges on each node diagram and otherwise disjoint to both node diagrams.

Connect the nodes by removing the two opposite edges and adding the other two edges of the rectangle.

Clear dismantling

A theorem of Schubert says that every node can be uniquely decomposed as a connected sum of prime nodes.

Indecomposability of diagrams

A node is prime if and only if each of its diagrams is prime. It is therefore sufficient to check indivisibility using a diagram.

Web links

• Eric W. Weisstein : Prime Knot . In: MathWorld (English).
• Eric W. Weisstein : Composite Knot . In: MathWorld (English).

Individual evidence

1. ↑ Horst Schubert : The clear decomposability of a node into prime nodes . S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), pp. 57-104.