For links, the writhe is not an invariant, because the change of orientation of one component induces a change of signs of all its crossings with other components. Therefore, we will define for links another invariant- the linking number. Let c1 and c2 be two components of link L. The linking number of those two components is the absolute value of the sum of signs of their crossing points divided by 2. The linking number of a link L is the sum of the linking numbers of all its components. For example, the linking number of Borromean rings is 0. The LinKnot function LinkingNo (webMathematica LinkingNo) calculates the linking number for any given link. 

Let's start with a KL shadow with n crossings. Then we take all possible combinations of signs +1 or -1 assigned to its vertices and obtain 2n states of the KL shadow. Each of them represents a projection of some KL. What will happen if we leave some vertices of a KL shadow unsigned? In this case one must consider singular links that differ from true KLs in that they possess double points where one part of the KL cuts another part transversally. Singular KLs are KLs with intersections. Their projections are called special projections and play a great role in the construction of Vassiliev invariants. Certainly, the number of all special projections that can be obtained from a KL shadow with n vertices is 3n

For any KL given by its Conway symbol, the LinKnot function fGenSign (webMathematica fGenSign) computes the signs of the crossing points in the order corresponding to the Dowker code or P-data of a given KL. The function fGaussExtSigns (webMathematica fGaussExtSigns) calculates the Gauss code with signs for a KL projection given by its Conway symbol, Dowker code, or P-data. The function fSignsKL (webMathematica fSignsKL) calculates the Dowker code with signs of a KL given by its Dowker code in Knotscape form (or DT-code). For an alternating knot, an input is the Dowker code without signs, and for a non-alternating KL, an input is the Dowker code containing only signs of points with signs changed with regard to the corresponding alternating KL. The output is a Dowker code with signs. The function fKnotscapeDow (webMathematica fKnotscapeDow) calculates from a Conway symbol of a KL its Dowker code in the Knotscape format: Dowker code without signs for an alternating KL, or Dowker code with signs of changed crossings for an non-alternating KL. 


PreviousContentsNext