The HOMFLY-PT Polynomial

KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

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

Failed to parse (unknown function "\overcrossing"): {\displaystyle 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[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 $a=q^{-1}$ and $z=q^{1/2}-q^{-1/2}$ and to the Conway polynomial at $a=1$. 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 }

8_1

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[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

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, $ConwayAlgebrasandSkeinEquivalenceofLinks$, Proc. Amer. Math. Soc. 100 (1987) 744-748.