A link is called prime if each of its shadows represents a graph which is at least 3-edge connected. In other words, a prime KL cannot be represented by some 2-edge connected shadow. A link which is not prime is called a composite link. The LinKnot function fPrimeKL (webMathematica fPrimeKL) checks if a KL is prime, giving as the output 1 for prime, and 0 for composite KLs. In a similar way, the function fPrimeGraph (webMathematica fPrimeGraph) tests whether a graph given by a list of unordered pairs, after transformation into the corresponding alternating KL, will represent a prime or composite alternating KL. The output is 1 for prime, and 0 for a composite KL

In the case of prime knots, the information contained in a DT-code is sufficient to draw the corresponding KL projection (or its mirror image), but ambiguity occurs in the case of alternating KLs that are not prime. For example, the two different composite knots possess the same DT-code {{6},{4,6,2,10,12,8}}. 

Introducing signs enables us to define a new invariant of alternating knots- the writhe. Writhe is defined as the sum of the signs of the crossing points of a knot. Usually, the invariance is checked by means of combinatorial moves, called Reidemeister moves, which will be given later. Writhe can be calculated by using Knot 2000 (K2K) function WritheKnotFromPdata (webMathematica WritheKnotFromPdata). For example, the writhe of the "right" trefoil knot is 3, the writhe of the left trefoil is -3, and the writhe of the composite knots is 6 and 0, respectively, so it is now easy to distinguish them. For alternating knots, writhe is an invariant. In the case of non-alternating knots, two different minimal projections of the same knot can have a different writhe. The first such example is the Perko pair

 

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