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I have two questions:

  1. From the definition of the Jones polynomial as the normalization of the Kauffman bracket $(-A^3)^{-w(D)} \langle D\rangle$ and substituting $A\rightarrow t^{-1/4}$, how does one obtain the skein relation for the Jones polynomial, $(t^{1/2}-t^{-1/2}) V_0(K)=t^{-1}V(K_+)-tV(K_-)$?

  2. If we take the definition of the HOMFLPT polynomial $lP(L_+)+l^{-1}P(L_-)+mP(L_0)=0$, how do I show that the Jones polynomial is a special case of the HOMFLYPT in the sence that $V(K_1) = V(K_2) \Rightarrow P(K_1)=P(K_2).$ If we take $m=-(t^{1/2}-t^{-1/2})$ and $l=t^{-1}$, the signs don't match (I know that we can take another definition where it works, but I'm wondering from this one).

Can the above two questions be answered using an elementary or combinatorial approach?

In addition, is there a reference for counterexamples that show that the HOMFLYPT is stronger than the Alexander, Jones, Kauffman?

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For 1, see Lickorish's "An Introduction to Knot Theory" Proposition 3.7 (page 28).

For 2, the substitution $l=i\alpha$ and $m=-i z$ gives the $\alpha P(L_+)-\alpha^{-1}P(L_-)=z P(L_0)$ version of the HOMFLY polynomial, from which $\alpha=t^{-1}$ and $z=t^{1/2}-t^{-1/2}$ gives the Jones polynomial.

For counterexamples, there is the Knotinfo database: http://www.indiana.edu/~knotinfo/

There is an analysis of the relative strengths at https://math.stackexchange.com/a/2767020/172988 for knots with up to twelve crossings:

  • The HOMFLY polynomial is determined by its Jones and Alexander-Conway polynomials.
  • There are knots distinguishable by their Jones polynomials but not their Alexander-Conway polynomials, and vice versa.
  • There are knots distinguishable by their Kauffman polynomials but not their HOMFLY polynomials, and vice versa.

According to https://mathoverflow.net/a/310027/43804 the first point does not hold in general.

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