The HOMFLY-PT polynomial is a knot and linkinvariant. Confusingly, there are several variants depending on exactly which relationships are used to define it. All are related by simple substitutions.
Definition
To compute the HOMFLY-PT polynomial, one starts from an oriented link diagram and uses the following rules:
Let $L_+$, $L_-$, and $L_0$ be links which are the same except for one part where they differ according to the diagrams below. Then, depending on the choice of variables:
P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, and A. Ocneanu. (1985). A New Polynomial Invariant of Knots and Links Bulletin of the American Mathematical Society 12 (2): 239–246.
More recent work includes:
A.Mironov, A.Morozov, An.Morozov, Character expansion for HOMFLY polynomials. I. Integrability and difference equations, arxiv/1112.5754
Hugh Morton, Peter Samuelson, The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra, arxiv/1410.0859