A Polynomial invariant of a Knot discovered in 1923 by J. W. Alexander (Alexander 1928). In technical language, the Alexander polynomial arises from the Homology of the infinitely cyclic cover of a Knot's complement. Any generator of a Principal Alexander Ideal is called an Alexander polynomial (Rolfsen 1976). Because the Alexander Invariant of a Tame Knot in has a Square presentation Matrix, its Alexander Ideal is Principal and it has an Alexander polynomial denoted .

Let be the Matrix Product of Braid Words of a Knot, then

(1) |

(2) |

(3) |

The Alexander polynomial remained the *only* known Knot Polynomial until the Jones Polynomial was
discovered in 1984. Unlike the Alexander polynomial, the more powerful Jones Polynomial *does,* in most cases,
distinguish Handedness. A normalized form of the Alexander polynomial symmetric in and and
satisfying

(4) |

(5) |

(6) | |||

(7) | |||

(8) |

for the Trefoil Knot, Figure-of-Eight Knot, and Solomon's Seal Knot, respectively. Multiplying through to clear the Negative Powers gives the usual Alexander polynomial, where the final Sign is determined by convention.

Let an Alexander polynomial be denoted , then there exists a Skein Relationship
(discovered by J. H. Conway)

(9) |

(10) |

For a Knot,

(11) |

(12) |

(13) |

The HOMFLY Polynomial generalizes the Alexander polynomial (as well at the Jones Polynomial)
with

(14) |

Rolfsen (1976) gives a tabulation of Alexander polynomials for Knots up to 10 Crossings and Links up to 9 Crossings.

**References**

Adams, C. C. *The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots.* New York:
W. H. Freeman, pp. 165-169, 1994.

Alexander, J. W. ``Topological Invariants of Knots and Links.'' *Trans. Amer. Math. Soc.* **30**, 275-306, 1928.

Alexander, J. W. ``A Lemma on a System of Knotted Curves." *Proc. Nat. Acad. Sci. USA* **9**, 93-95, 1923.

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. Amer. Math. Soc.* **12**, 103-111, 1985.

Rolfsen, D. ``Table of Knots and Links.'' Appendix C in *Knots and Links.* Wilmington, DE:
Publish or Perish Press, pp. 280-287, 1976.

Stoimenow, A. ``Alexander Polynomials.'' http://www.informatik.hu-berlin.de/~stoimeno/ptab/a10.html.

Stoimenow, A. ``Conway Polynomials.'' http://www.informatik.hu-berlin.de/~stoimeno/ptab/c10.html.

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1999-05-25