1.10 Prime and composite KLsThe main property of the program LinKnot is the opportunity to make experiments with KLs given in a comprehensive way in Conway notation, and try to discover some still undiscovered regularities. At first glance, KLs might seem to be an unordered, random structure, similar to that of prime numbers. Indeed, there is some analogy between KLs and prime numbers: in knot theory we also have prime KLs. Again by a KL surgery, given any two links L_{1} and L_{2} we can define their composition (connected sum, or direct product) denoted by L_{1}#L_{2}, as illustrated in the corresponding figure. Suppose that a sphere in Â^{3}intersects a link L in exactly two points. This splits a link L into two arcs. The endpoints of either of those arcs can be joined by an arc lying on the sphere. That construction results in two links, L_{1} and L_{2}. The links L_{1} and L_{2} that make up the composite link L are called factor links (or simply, factors). A link is called prime if in every decomposition into a connected sum, one of the factors is unknotted. For prime knots the following properties hold:
The prime decomposition theorem (proven by H. Schubert in 1949) holds for links: Theorem If L is a link, then there is a decomposition L = L_{1}#...#L_{m} where each L_{i} (i = 1,...,m) is a prime link. Such a decomposition is unique up to order.
