---
tag: QBQ0BH589C
kind: concept
name: HOMFLY polynomial
msc: [57K14]
specializes: [GZQ5ZE3X6G]
sources:
  - url: https://en.wikipedia.org/wiki/HOMFLY_polynomial
same_as:
  wikidata: Q996437
  wikipedia: HOMFLY polynomial
---

## Summary

The HOMFLY polynomial is an invariant of oriented [[links|1A5WRCNTM3]] with values in Laurent polynomials in two variables.
It is defined by a skein relation and contains both the [[Alexander polynomial|JKA25KDPQP]] and the [[Jones polynomial|4SPC17FPXE]] as special cases.
Its name comes from the initials of Hoste, Ocneanu, Millett, Freyd, Lickorish and Yetter, who found it soon after the Jones polynomial.
Przytycki and Traczyk found it independently, so it is also called the HOMFLYPT polynomial.

## Explanations

### L1 human

The HOMFLY polynomial is a two-variable polynomial attached to a knot or link.
Like the Jones polynomial, you compute it by undoing crossings one at a time with a fixed rule.
Setting the two variables to special values gives back older invariants.
So one calculation gives both the Alexander and the Jones polynomial at once.