© | Dror Bar-Natan: Knot Atlas: KnotTheory`:               This page is passe. Go here instead!

7.8 The HOMFLY-PT Polynomial

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

$$aH\left(\text{\LARGE\raisebox{-4pt}{$\overcrossing$}}\right) -a^{... ...crossing$}}\right) = zH\left(\text{\LARGE\raisebox{-4pt}{$\smoothing$}}\right)$$

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

KnotTheory` knows about the HOMFLY-PT polynomial:

In[1]

In[2]:= ?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[3]:= 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[4]:=

K = Knot[8, 1];

In[5]:=	HOMFLYPT[Knot[8, 1]][a, z]
Out[5]=	-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[6]:=	{Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]], Jones[K][q]}
Out[6]=	-6 -5 -4 2 2 2 2 {2 + q - q + q - -- + -- - - - q + q , 3 2 q q q -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