HOMFLY polynomial


In the mathematical field of knot theory, the 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 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 and 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, and the Alexander polynomial, can be computed in terms of the HOMFLY polynomial as follows: