“Alexander Polynomial”的概念、定义、习题解题方法及翻译-数学辞典

单词

Alexander Polynomial

释义

Alexander Polynomial

A Polynomial invariant of a Knot discovered in 1923 by J. W. Alexander (Alexander 1928). In technicallanguage, the Alexander polynomial arises from the Homology of the infinitely cyclic coverof a Knot's complement. Any generator of a Principal Alexander Ideal is called anAlexander polynomial (Rolfsen 1976). Because the Alexander Invariant of a Tame Knot in has aSquare 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)

where

is the Alexander polynomial and det is the Determinant. The Alexander polynomial of a TameKnot in

satisfies

(2)

where

is a Seifert Matrix, det is the Determinant, and

denotes the Matrix Transpose. The Alexander polynomial also satisfies

(3)

The Alexander polynomial of a splittable link is always 0. Surprisingly, there are known examples of nontrivialKnots with Alexander polynomial 1. An example is the

Pretzel Knot.

The Alexander polynomial remained the only known Knot Polynomial until the Jones Polynomial wasdiscovered 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 andsatisfying

(4)

was formulated by J. H. Conway and is sometimes denoted

. The Notation

is anabbreviation for the Conway-normalized Alexander polynomial of a Knot

(5)

For a description of the Notation for Links, see Rolfsen (1976, p. 389). Examples ofthe Conway-Alexander polynomials for common Knots include

			(6)
			(7)
			(8)

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

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

(9)

corresponding to the above Link Diagrams (Adams 1994). A slightly different SkeinRelationship convention used by Doll and Hoste (1991) is

(10)

These relations allow Alexander polynomials to be constructed for arbitrary knots by building them up as a sequenceof over- and undercrossings.

For a Knot,

(11)

where Arf is the Arf Invariant (Jones 1985). If

is a Knot and

(12)

then

cannot be represented as a closed 3-Braid. Also, if

(13)

then

cannot be represented as a closed 4-braid (Jones 1985).

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

(14)

(Doll and Hoste 1991).

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

See also Braid Group, Jones Polynomial, Knot, Knot Determinant,Link, Skein Relationship
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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