A 2-variable oriented Knot Polynomial motivated by the Jones Polynomial (Freyd *et al. *1985). Its name
is an acronym for the last names of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter (Freyd *et al. *
1985). Independent work related to the HOMFLY polynomial was also carried out by Prztycki and Traczyk (1987). HOMFLY
polynomial is defined by the Skein Relationship

(1) |

(2) |

(3) |

(4) |

It is normalized so that
. Also, for unlinked unknotted components,

(5) |

(6) | |||

(7) |

It is also a generalization of the Alexander Polynomial , satisfying

(8) |

(9) |

A split union of two links (i.e., bringing two links together without intertwining them) has HOMFLY polynomial

(10) |

(11) |

Mutants have the same HOMFLY polynomials. In fact, there are infinitely many distinct Knots with the same HOMFLY Polynomial (Kanenobu 1986). Examples include (05-001, 10-132), (08-008, 10-129) (08-016, 10-156), and (10-025, 10-056) (Jones 1987). Incidentally, these also have the same Jones Polynomial.

M. B. Thistlethwaite has tabulated the HOMFLY polynomial for Knots up to 13 crossings.

**References**

Adams, C. C. *The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots.* New York:
W. H. Freeman, pp. 171-172, 1994.

Doll, H. and Hoste, J. ``A Tabulation of Oriented Links.'' *Math. Comput.* **57**, 747-761, 1991.

Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millett, K.; and Oceanu, A. ``A New Polynomial Invariant of
Knots and Links.'' *Bull. Amer. Math. Soc.* **12**, 239-246, 1985.

Jones, V. ``Hecke Algebra Representations of Braid Groups and Link Polynomials.'' *Ann. Math.* **126**, 335-388, 1987.

Kanenobu, T. ``Infinitely Many Knots with the Same Polynomial.'' *Proc. Amer. Math. Soc.* **97**, 158-161, 1986.

Kanenobu, T. and Sumi, T. ``Polynomial Invariants of 2-Bridge Knots through 22 Crossings.'' *Math. Comput.* **60**, 771-778 and S17-S28, 1993.

Kauffman, L. H. *Knots and Physics.* Singapore: World Scientific, p. 52, 1991.

Lickorish, W. B. R. and Millett, B. R. ``The New Polynomial Invariants of Knots and Links.'' *Math. Mag.* **61**, 1-23, 1988.

Morton, H. R. and Short, H. B. ``Calculating the -Variable Polynomial for Knots Presented as Closed Braids.''
*J. Algorithms* **11**, 117-131, 1990.

Przytycki, J. and Traczyk, P. ``Conway Algebras and Skein Equivalence of Links.'' *Proc. Amer. Math. Soc.*
**100**, 744-748, 1987.

Stoimenow, A. ``Jones Polynomials.'' http://www.informatik.hu-berlin.de/~stoimeno/ptab/j10.html.

Weisstein, E. W. ``Knots and Links.'' Mathematica notebook Knots.m.

© 1996-9

1999-05-25