# The HOMFLY-PT Polynomial

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

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

and by the initial condition ${\displaystyle H(\bigcirc )}$=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 ${\displaystyle a=q^{-1}}$ and ${\displaystyle z=q^{1/2}-q^{-1/2}}$ and to the Conway polynomial at ${\displaystyle a=1}$. 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 L5a1

In our parametrization of the ${\displaystyle A_{2}}$ link invariant, it satisfies

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

where ${\displaystyle L}$ is some knot or link and where ${\displaystyle c}$ is the number of components of ${\displaystyle L}$. 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.