The second Knot Polynomial discovered. Unlike the first-discovered Alexander Polynomial, the Jones polynomial
can sometimes distinguish handedness (as can its more powerful generalization, the HOMFLY Polynomial). Jones
polynomials are Laurent Polynomials in assigned to an Knot. The Jones
polynomials are denoted for Links, for Knots, and normalized so that

(1) |

(2) |

If a Link has an Odd number of components, then is a Laurent Polynomial over the
Integers; if the number of components is Even, is times a Laurent Polynomial.
The Jones polynomial of a Knot Sum satisfies

(3) |

The Skein Relationship for under- and overcrossings is

(4) |

Some interesting identities from Jones (1985) follow. For any Link ,

(5) |

(6) |

(7) |

(8) |

Let denote the Mirror Image of a Knot . Then

(9) |

(10) | |||

(11) |

Jones defined a simplified trace invariant for knots by

(12) |

(13) |

Jones polynomials were subsequently generalized to the two-variable HOMFLY Polynomials, the
relationship being

(14) |

(15) |

(16) |

(17) |

(18) |

Birman and Lin (1993) showed that substituting the Power Series for as the variable in the Jones polynomial yields a Power Series whose Coefficients are Vassiliev Polynomials.

Let be an oriented connected Link projection of crossings, then

(19) |

There exist distinct Knots with the same Jones polynomial. Examples include (05-001, 10-132), (08-008, 10-129), (08-016, 10-156), (10-025, 10-056), (10-022, 10-035), (10-041, 10-094), (10-043, 10-091), (10-059, 10-106), (10-060, 10-083), (10-071, 10-104), (10-073, 10-086), (10-081, 10-109), and (10-137, 10-155) (Jones 1987). Incidentally, the first four of these also have the same HOMFLY Polynomial.

Witten (1989) gave a heuristic definition in terms of a topological quantum field theory, and Sawin (1996) showed that the ``quantum group'' gives rise to the Jones polynomial.

**References**

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

Birman, J. S. and Lin, X.-S. ``Knot Polynomials and Vassiliev's Invariants.'' *Invent. Math.* **111**, 225-270, 1993.

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

Jones, V. ``A Polynomial Invariant for Knots via von Neumann Algebras.'' *Bull. Am. Math. Soc.* **12**, 103-111, 1985.

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

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

Murasugi, K. ``Jones Polynomials and Classical Conjectures in Knot Theory.'' *Topology* **26**, 297-307, 1987.

Praslov, V. V. and Sossinsky, A. B.
*Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology.*
Providence, RI: Amer. Math. Soc., 1996.

Sawin, S. ``Links, Quantum Groups, and TQFTS.'' *Bull. Amer. Math. Soc.* **33**, 413-445, 1996.

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

Thistlethwaite, M. ``A Spanning Tree Expansion for the Jones Polynomial.'' *Topology* **26**, 297-309, 1987.

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

Witten, E. ``Quantum Field Theory and the Jones Polynomial.'' *Comm. Math. Phys.* **121**, 351-399, 1989.

© 1996-9

1999-05-25