The HOMFLY-PT Polynomial
From Knot Atlas
and by the initial condition =1.
KnotTheory` knows about the HOMFLY-PT polynomial:
(For In see Setup)
Thus, for example, here's the HOMFLY-PT polynomial of the knot 8_1:
It is well known that HOMFLY-PT polynomial specializes to the Jones polynomial at a = q−1 and z = q1 / 2−q−1 / 2 and to the Conway polynomial at a = 1. Indeed,
In our parametrization of the A2 link invariant, it satisfies
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:
 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 .
[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.