2.3  Vassiliev invariants 

The universal invariants from which many other KL-invariants can be deduced are Vassiliev invariants (or finite-order invariants). In a crossing change from an overcrossing to an undercrossing Vassiliev introduced an intermediate phase: a catastrophe, when one part of a KL cuts another part transversely. As the result, we obtain singular knots or links. Their projections contain, besides overcrossings and undercrossings, double points where KL cuts itself. If we denote the set of all singular KLs by F, ordinary KLs without special crossings form a subset of F denoted by S0. In the same sense, the remaining part of the set F can be subdivided into strata S1, S2, S3,... consisting of singular KLs with 1,2,3,... double points, respectively. In every crossing change from overcrossing to undercrossing or vice versa, a KL becomes singular, passing through an intermediate phase- a catastrophe. In the same way as before we can define an ambient isotopy for singular KLs: two (singular) knots or links L1 and L2 are isotopic if there is an orientation-preserving homeomorphism of Â3 that sends L1 to L2 preserving the arrows indicating orientation and the cyclic order of the branches with double points. Working with special projections, in order to establish the equivalence of singular KLs, together with Reidemeister moves defined before for ordinary KLs, near special vertices we introduce the operation W. As an equivalent of a skein relation for singular KLs we introduce the following relation and define the derivative v' of the invariant v as