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

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

KnotTheory knows about the HOMFLY-PT polynomial:

(For In see Setup)

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

In our parametrization 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:= L = Link[5, Alternating, 1];
 In:= Simplify[{ (-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q], A2Invariant[L][q] }] Out=  -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`

#### Other Software to Compute the HOMFLY-PT Polynomial

A C-based program running under windows by M. Ochiai can compute the HOMFLY-PT polynomial of certain knots and links with up to hundreds of crossings using "base tangle decompositions". His program, bTd, is available at .