HOMFLYPT is an acronym for the six researchers working in four different independent groups, who discovered the same polynomial at the same time and published their results simultaneously (in 1985) in the same journal. The name HOMFLYPT comes from the first letters of the names of the discoverers: Hoste, Ocneau, Millett, Freyd, Lickorish, and Yetter. J. The first six of them published their papers together in 1985 (Freyd, Yetter, Hoste, Lickorish, Millett, Onceanu, 1985), and J. Przytycki and P. Traczyk published their work somewhat later, because it didn't arrive by mail on time. The simplest way to define the HOMFLYPT polynomial P(l,m) with two variables l and m is to use the Conway rules with P in the place of and with the modified skein relations 

lP(D+)-mP(D-) = P(D0).

The HOMFLYPT polynomial (or two-variable Jones polynomial) is more sensitive than the Conway polynomial, but it is still an incomplete invariant, unable to distinguish all non-isotopic knots. For example, for two different eleven-crossing mutant knots .2.(3,2) and .2.(2,3) the Alexander polynomials, Jones polynomials, HOMFLYPT polynomials, and Kauffman polynomials are all equal. Any pair of alternating KLs can easily be distinguished using the LinKnot function SameAltConKL (webMathematica SameAltConKL) that compares minimal Dowker codes. For example, the minimal Dowker code of the knot .2.(3,2) is 

{{11}, {6, 8, -12, 2, 20, -18, -4, -10, 22, -14,16}},
and the minimal Dowker code of the knot .2.(2,3) is 
{{11}, {6, 8, -12, 2, -18, 16, -4, 20, -22, 14,-10}}.