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

Failed to parse (unknown function "\overcrossing"): {\displaystyle 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)

In[1]:= ?HOMFLYPT HOMFLYPT[K][a, z] computes the HOMFLY-PT (Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Przytycki and Traczyk) polynomial of a knot/link K, in the variables a and z.

In[2]:= HOMFLYPT::about The HOMFLYPT program was written by Scott Morrison.

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

In[3]:=

K = Knot[8, 1];

In[4]:=	HOMFLYPT[Knot[8, 1]][a, z]
Out[4]=	-2 4 6 2 2 2 4 2 a - a + a - z - a z - a z

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[5]:=	Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]
Out[5]=	-6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q

In[6]:=	Jones[K][q]
Out[6]=	-6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q

In[7]:=	{HOMFLYPT[K][1, z], Conway[K][z]}
Out[7]=	2 2 {1 - 3 z , 1 - 3 z }

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:

In[8]:=

L = Link[5, Alternating, 1];

In[9]:=	Simplify[{ (-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q], A2Invariant[L][q] }]
Out[9]=	-12 -8 -6 2 -2 2 4 6 {2 - q + q + q + -- + q + q + q + q , 4 q -12 -8 -6 2 -2 2 4 6 2 - q + q + q + -- + q + q + q + q } 4 q

[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.