HOMFLY polynomial

In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables "m" and "l". 
A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an invariant of the knot, i.e. diagrams representing the same knot have the same polynomial. The converse may not be true. The HOMFLY polynomial is one such invariant and it generalizes two polynomials previously discovered, the Alexander polynomial and the Jones polynomial, both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a quantum invariant.
The name "HOMFLY" combines the initials of its co-discoverers: Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter J. Freyd, W. B. R. Lickorish, and David N. Yetter. The addition of "PT" recognizes independent work carried out by Józef H. Przytycki and Paweł Traczyk.
Definition.
The polynomial is defined using skein relations:
where formula_3 are links formed by crossing and smoothing changes on a local region of a link diagram, as indicated in the figure. 
The HOMFLY polynomial of a link "L" that is a split union of two links formula_4 and formula_5 is given by
See the page on skein relation for an example of a computation using such relations.
Other HOMFLY skein relations.
This polynomial can be obtained also using other skein relations:
Main properties.
The Jones polynomial, "V"("t"), and the Alexander polynomial, formula_11 can be computed in terms of the HOMFLY polynomial (the version in formula_12 and formula_13 variables) as follows: