Unfortunately, the main problem is the second step: reduction of KLs is not always successful, so we are not sure that we have obtained a minimal projection of the KL in question. To make sure that the KL considered is really a new non-alternating KL with n crossings (and not an alternating or non-alternating KL with a lower number of crossings, or an already derived non-alternating KL with n crossings) we need to compare its polynomial invariants with the polynomial invariants of all KLs with a lower number of crossings and non-alternating KLs with n crossings. To speed up the process, it is useful to subdivide KLs according to the number of components and work inside the corresponding classes (knots, 2-component links, 3-component links, etc.) We already mentioned that the equality of polynomial invariants is not a complete guarantee that two KLs are equal. For greater assurance that we have not eliminated some non-alternating KLs that are really new, since that fact cannot be recognized by their polynomial invariants, we can compute all polynomial invariants (and even other invariants) for the KL in question and compare them with the corresponding invariants of all KLs previously derived.

1. Some reduction procedures are used in the initial generation of KLs, and these constitute the first filter;
 
2. Polynomials are used at the next stage - they can be quickly computed, although they are not very powerful at distinguishing KLs. However, they do resolve the KLs into relatively small equivalence classes;
 
3. Now we need more powerful invariants, like homomorphisms of the KL group or those arising from hyperbolic structure (if it exists). Finally, one has to work hard to show that the groups of diagrams that haven't been distinguished so far, actually do represent the same KL.

Another more promising possibility is the symmetry-oriented and graph-theoretical approach, used by J. Conway (1970) and A. Caudron (1982), but its computer implementation is rather difficult. In this approach we try to establish some hierarchy of all different representations of an non-alternating KLs given in Conway notation (in a similar way as in the case of other worlds already considered) and to find an algorithmic approach which will enable us to produce a complete list of non-alternating KLs without duplications (or, at least, with a minimal number of repetitions that could be eliminated by polynomial or other invariants).