The HOMFLY-PT Polynomial

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The HOMFLY-PT polynomial $H(L)(a,z)$ (see [HOMFLY] and [PT] of a knot or link $L$ is defined by the skein relation

$aH\left(\{overcrossing\}\right)-a^{-1}H\left(\{undercrossing\}\right)=zH\left(\{smoothing\}\right)$

and by the initial condition $H(\{bigcirc\})$=1.

KnotTheory` knows about the HOMFLY-PT polynomial:

(For In[1] see Setup)

Thus, for example, here's the HOMFLY-PT polynomial of the knot 8_1:

It is well known that HOMFLY-PT polynomial specializes to the Jones polynomial at $a=q^{-1}$ and $z=q^{1/2}-q^{-1/2}$ and to the Conway polynomial at $a=1$. Indeed,

In our parametirzation of the $A_{2}$ link invariant, it satisfies

$A_{2}(L)(q)=(-1)^{c}(q^{2}+1+q^{-2})H(L)(q^{-3},\,q-q^{-1})$,

where $L$ is some knot or link and where $c$ is the number of components of $L$. Let us verify this fact for the Whitehead link, L5a1:

[HOMFLY] ^  J. Hoste, A. Ocneanu, K. Millett, P. Freyd, W. B. R. Lickorish and D. Yetter, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 239-246.

[PT] ^  J. Przytycki and P. Traczyk, $ConwayAlgebrasandSkeinEquivalenceofLinks$, Proc. Amer. Math. Soc. 100 (1987) 744-748.