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)

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]]]
Out[6]= -6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[7]:= Jones[K][q]
Out[7]= -6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[8]:= {HOMFLYPT[K][1, z], Conway[K][z]}
Out[8]= 2 2 {1 - 3 z , 1 - 3 z }
8 1.gif
8_1
L5a1.gif
L5a1

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[9]:= L = Link[5, Alternating, 1];
In[10]:= Simplify[{ (-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q], A2Invariant[L][q] }]
Out[10]= -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 [1].

References

[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, Conway Algebras and Skein Equivalence of Links, Proc. Amer. Math. Soc. 100 (1987) 744-748.