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.
- 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?
This answer is purely empirical, using the knotinfo database for all knots up through $12$ crossings.
$4_1$ 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.)