For every polynomial we can define its coefficient of efficiency in distinguishing KLs: the number of different KLs that it can recognize as different, divided by the number of all really different KLs with a given number of crossings n. However, for non-alternating KLs, the exactness of the second number is impossible to prove by using polynomials: two really different KLs (e.g., mutant KLs) can have all existing polynomial invariants in common. This is more likely to happen then one might expect, because some polynomial invariants are only special cases of some more general invariants. For example, Alexander polynomials can be obtained from Conway polynomials or Jones polynomials. 

It seems that for every polynomial KL invariant it is possible to construct infinite classes of different KLs that this invariant is not able to distinguish. For example, all KLs obtained from a KL of the form p1,p2,...,pn by any permutation of tangles p1, p2, ..., pn cannot be distinguished by the Kauffman polynomial. 

An interesting class of KLs are those with trivial Alexander and Conway polynomials. We assume that every link with a trivial Alexander and Conway polynomial is a member of an infinite family of links, all having trivial Alexander and Conway polynomials. 

For n = 8 the 4-component link 2,2,-2,-2 is a member of the family 2k,2k,-2k,-2k; for n = 9 the 3-component link .-(2,2) that has the non-minimal algebraic 10-crossing representation (2,-2),(-2,2),2 is a member of the family (2k,-2k),(-2m,2m),2n with trivial Alexander and Conway polynomials. For n = 10: 

the 3-component link (3,2 1) -(2,2) is a member of the family (2k+1,2k 1) -(2,2); 

the 3-component link 3,2 1,-2,-2 is a member of the family 2k+1,2k 1,-2,-2; 

the 3-component link 3,-2,2 1,-2 is a member of the family 2k+1,-2,2k 1,-2; 

the 4-component link (2,2) -2 (2,2) that has the non-minimal 12-crossing representation (2,2-) 2 (2,2-) is a member of the family (2,2-) 2k (2,2-); 

the 4-component link .-(2,2):2 that has the non-minimal 12-crossing algebraic representation (2,2) -1,3,(2,-2) is a member of the family (2,2) -1,2k+1,(2,-2); 

the 4-component link 103*-1.-1.-1.-1::.-1 that has the non-minimal 12-crossing algebraic representation (2,-2),-2,2,(2,-2) is a member of the family (2,-2),-2k,2k,(2,-2) with trivial Alexander and Conway polynomials. 

Hence, for n £10 we succeeded to show that all links with trivial Alexander and Conway polynomials, except one, are members of the infinite families having the same property. The only exception is the link 2 0.-2.-2 0.2 0 for which we have not succeeded to find the corresponding family of links with trivial Alexander and Conway polynomials. 

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