The HOMFLY-PT Polynomial

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

[math]\displaystyle{ aH\left(\{overcrossing\}\right) -a^{-1}H\left(\{undercrossing\}\right) = zH\left(\{smoothing\}\right) }[/math]

and by the initial condition [math]\displaystyle{ H(\{bigcirc\}) }[/math]=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 [math]\displaystyle{ a=q^{-1} }[/math] and [math]\displaystyle{ z=q^{1/2}-q^{-1/2} }[/math] and to the Conway polynomial at [math]\displaystyle{ a=1 }[/math]. 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 [math]\displaystyle{ A_2 }[/math] link invariant, it satisfies

[math]\displaystyle{ A_2(L)(q) = (-1)^c(q^2+1+q^{-2})H(L)(q^{-3},\,q-q^{-1}) }[/math],

where [math]\displaystyle{ L }[/math] is some knot or link and where [math]\displaystyle{ c }[/math] is the number of components of [math]\displaystyle{ L }[/math]. 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 -12 -8 -6 2 -2 2 4 6 {2 - q + q + q + -- + q + q + q + q , 2 - q + q + q + -- + q + q + q + q } 4 4 q 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, [math]\displaystyle{ Conway Algebras and Skein Equivalence of Links }[/math], Proc. Amer. Math. Soc. 100 (1987) 744-748.