Knot polynomial

Knot polynomial

In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.

History

The first knot polynomial, the Alexander polynomial, was introduced by J. W. Alexander in 1923, but other knot polynomials were not found until almost 60 years later.

In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander-Conway polynomial. The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial.

Soon after Jones' discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a state-sum model, which involved the bracket polynomial, an invariant of framed knots. This opened up avenues of research linking knot theory and statistical mechanics.

In the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern-Simons theory. Viktor Vassiliev and Mikhail Goussarov started the theory of finite type invariants of knots. The coefficients of the previously named polynomials are known to be of finite type (after perhaps a suitable "change of variables").

In recent years, the Alexander polynomial has been shown to be related to Floer homology. The graded Euler characteristic of the knot Floer homology of Oszvath and Szabo is the Alexander polynomial.

References

*Colin Adams, "The Knot Book", American Mathematical Society, ISBN 0-8050-7380-9
*W. B. R. Lickorish, "An introduction to knot theory." Graduate Texts in Mathematics, 175. Springer-Verlag, New York, 1997. ISBN 0-387-98254-X

ee also

Specific knot polynomials

*Alexander polynomial
*Bracket polynomial
*HOMFLY polynomial
*Jones polynomial
*Kauffman polynomial

Related topics

*skein relationship for a formal definition of the Alexander polynomial, with a worked-out example.


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Knot theory — A three dimensional depiction of a thickened trefoil knot, the simplest non trivial knot …   Wikipedia

  • Knot invariant — In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some… …   Wikipedia

  • List of knot theory topics — Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician s knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical… …   Wikipedia

  • Knot — For speeds, see Knot (unit). For other uses, see Knot (disambiguation). Some knots 1 Yarn knot ABoK #2688 2 Manrope knot 3 Granny knot 4 Rosebud stopper knot(?) …   Wikipedia

  • Alexander polynomial — In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a… …   Wikipedia

  • Jones polynomial — In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1983. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial …   Wikipedia

  • HOMFLY polynomial — In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY PT polynomial or the generalized Jones polynomial, is a 2 variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and… …   Wikipedia

  • Kauffman polynomial — The Kauffman polynomial is a 2 variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as:F(K)(a,z)=a^{ w(K)}L(K),where w(K) is the writhe of the link diagram and L(K) is a polynomial in a and z defined on… …   Wikipedia

  • List of polynomial topics — This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics.Basics*Polynomial *Coefficient *Monomial *Polynomial long division *Polynomial factorization *Rational function *Partial… …   Wikipedia

  • History of knot theory — For thousands of years, knots have been used for basic purposes such as recording information, fastening and tying objects together. Over time people realized that different knots were better at different tasks, such as climbing or sailing. Knots …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”