Classification of polynomial knot invariants? I am trying to find some information regarding the differences between the following knot invariants: Conway, Jones, Kauffman and HOMFLY.
I know that HOMFLY can be reduced to Conway and Jones. And I know that Kauffman can also be reduced to Jones. I want to know what is the difference between each. For example I know Jones can detect chirality but Conway cannot. 


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*Is there something that Conway can detect that Jones cannot? 

*What makes Kauffman more powerful than Jones and Conway?

*What makes HOMFLY more powerful than Jones and Conway?

*Which ambient isotopy classes of links are detected only by Kauffman and not by HOMFLY?

*Which ambient isotopy classes of links are detected only by HOMFLY and not by Kauffman?


Where can I find this sort of information?
 A: This answer is purely empirical, using the knotinfo database for all knots up through $12$ crossings.


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*$4_1$ and $11\mathrm{n}_{19}$ have the same Jones polynomials but different Alexander-Conway polynomials. (This is but one of $250$ such Jones polynomials.) Conversely, there are $561$ Alexander-Conway polynomials with knots that have different Jones polynomials (an interesting case being knots with trivial Alexander-Conway polynomial).

*I don't think the Kauffman polynomial is any more or less powerful than the Jones and Alexander-Conway polynomials.  There are four Kauffman polynomials with knots that are distinguishable by their Jones and Alexander-Conway polynomials, for example $12\mathrm{a}_{0301}$ vs $12\mathrm{a}_{0351}$. Conversely there are $128$, for example $7_1$ and $12\mathrm{n}_{0749}$ have the same Jones and Alexander-Conway polynomials, but are distinguishable by the Kauffman polynomial.

*For all the knots, the Jones and Alexander-Conway polynomials determine the HOMFLY polynomial. (I do not know if this holds in general.)
