HOMFLYPT

The HOMFLYPT polynomial, also called the HOMFLY-PT polynomial, is a two-variable Laurent polynomial invariant of oriented links. It generalizes both the Alexander polynomial and the Jones polynomial and plays a central role in knot theory. The invariant is named after several researchers who contributed to its discovery and development: Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, and Przytycki (the acronym is sometimes extended to include Traczyk). It can be defined via skein relations or via quantum-group methods.

For an oriented link L, the HOMFLYPT polynomial P_L(a,z) is the Laurent polynomial in two variables defined by P_{unknot} = 1 and the skein relation a P_{L+} - a^{-1} P_{L-} = z P_{L0}, where L+, L-, and L0 denote three oriented link diagrams that are identical except at one crossing, where they differ as in the standard skein triple. The polynomial is invariant under ambient isotopy and therefore an invariant of the oriented link.

It has useful specialization properties: by choosing particular values of the variables, one recovers other classical invariants. In particular, appropriate specializations yield the Alexander-Conway polynomial and the Jones polynomial. More generally, the HOMFLYPT polynomial encodes the family of quantum-group invariants associated with sl(n) and appears as a unifying framework for various polynomial link invariants. It is computable by repeated application of the skein relation or via state-sum models.

History and significance: The HOMFLYPT polynomial unifies several knot invariants and provides a powerful tool for distinguishing links. It is a standard object of study in knot theory, with connections to representation theory, categorification, and topological quantum field theory. The acronym reflects the researchers who independently discovered the invariant.