1.10  Prime and composite KLs 

The 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 L1 and L2 we can define their composition (connected sum, or direct product) denoted by L1#L2, as illustrated in the corresponding figure. Suppose that a sphere in Â3intersects 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, L1 and L2. The links L1 and L2 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: 

  1. if K1 = K2, then for any K, K1#K = K2#K; 
  2. for any K1, K2, K1#K2 = K2#K1 (commutativity); 
  3. for any K1, K2, K3, (K1#K2)#K3 = K1#(K2#K3) (associativity); 
  4. for any K, K#1 = K, where 1 is an unknot (neutral element). 

  5.  

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 = L1#...#Lm where each Li (i = 1,...,m) is a prime link. Such a decomposition is unique up to order. 

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