To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

HOMFLY polynomial

From Wikipedia, the free encyclopedia

In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or 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.[1] The addition of PT recognizes independent work carried out by Józef H. Przytycki and Paweł Traczyk.

YouTube Encyclopedic

  • 1/3
    1 581
    2 421
    13 089
  • Introduction to the Alexander Polynomial
  • Introduction to the Jones Polynomial
  • AlgTop22: Knots and surfaces I




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

, where # denotes the knot sum; thus the HOMFLY polynomial of a composite knot is the product of the HOMFLY polynomials of its components.
, so the HOMFLY polynomial can often be used to distinguish between two knots of different chirality. However there exist chiral pairs of knots that have the same HOMFLY polynomial, e.g. knots 942 and 1071[2]

The Jones polynomial, V(t), and the Alexander polynomial, can be computed in terms of the HOMFLY polynomial (the version in and variables) as follows:


  1. ^ Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., and Ocneanu, A. (1985). "A New Polynomial Invariant of Knots and Links". Bulletin of the American Mathematical Society. 12 (2): 239–246. doi:10.1090/S0273-0979-1985-15361-3.
  2. ^

Further reading

External links

This page was last edited on 26 January 2018, at 17:14
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.