To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Knot polynomial

## From Wikipedia, the free encyclopedia

Many knot polynomials are computed using skein relations, which allow one to change the different crossings of a knot to get simpler knots.

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.

### YouTube Encyclopedic

• 1/5
Views:
13 245
2 473
1 631
1 166
1 177
• AlgTop22: Knots and surfaces I
• Introduction to the Jones Polynomial
• Introduction to the Alexander Polynomial
• Introduction to knots
• Introduction to Diagrams and Reidemeister Moves

## History

The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II 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 Ozsváth and Szabó is the Alexander polynomial.

## Example

Alexander–Briggs notation Alexander polynomial ${\displaystyle \Delta (t)}$ Conway polynomial ${\displaystyle \nabla (z)}$ Jones polynomial ${\displaystyle V(q)}$ HOMFLY polynomial ${\displaystyle H(a,z)}$
${\displaystyle 0_{1}}$ (Unknot) ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
${\displaystyle 3_{1}}$ (Trefoil Knot) ${\displaystyle t-1+t^{-1}}$ ${\displaystyle z^{2}+1}$ ${\displaystyle q^{-1}+q^{-3}-q^{-4}}$ ${\displaystyle -a^{4}+a^{2}z^{2}+2a^{2}}$
${\displaystyle 4_{1}}$ (Figure-eight Knot) ${\displaystyle -t+3-t^{-1}}$ ${\displaystyle -z^{2}+1}$ ${\displaystyle q^{2}-q+1-q^{-1}+q^{-2}}$ ${\displaystyle a^{2}+a^{-2}-z^{2}-1}$
${\displaystyle 5_{1}}$ (Cinquefoil Knot) ${\displaystyle t^{2}-t+1-t^{-1}+t^{-2}}$ ${\displaystyle z^{4}+3z^{2}+1}$ ${\displaystyle q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}}$ ${\displaystyle -a^{6}z^{2}-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}}$
${\displaystyle -}$ (Granny Knot) ${\displaystyle \left(t-1+t^{-1}\right)^{2}}$ ${\displaystyle \left(z^{2}+1\right)^{2}}$ ${\displaystyle \left(q^{-1}+q^{-3}-q^{-4}\right)^{2}}$ ${\displaystyle \left(-a^{4}+a^{2}z^{2}+2a^{2}\right)^{2}}$
${\displaystyle -}$ (Square Knot) ${\displaystyle \left(t-1+t^{-1}\right)^{2}}$ ${\displaystyle \left(z^{2}+1\right)^{2}}$ ${\displaystyle \left(q^{-1}+q^{-3}-q^{-4}\right)\left(q+q^{3}-q^{4}\right)}$ ${\displaystyle \left(-a^{4}+a^{2}z^{2}+2a^{2}\right)\times }$
${\displaystyle \left(-a^{-4}+a^{-2}z^{-2}+2a^{-2}\right)}$

Alexander–Briggs notation is a notation that simply organizes knots by their crossing number. The order of Alexander–Briggs notation of prime knot is usually sured. (See List of prime knots.)

Notice that Alexander polynomial and Conway polynomial can not recognize the difference of left-trefoil knot and right-trefoil knot.

So the same situation as granny knot and square knot,since the addition of knots in ${\displaystyle \mathbb {R} ^{3}}$ is the product of knots in knot polynomials.

## See also

### Related topics

• Graph polynomial, a similar class of polynomial invariants in graph theory
• Skein relation for a formal definition of the Alexander polynomial, with a worked-out example.

## Further reading

• Adams, Colin. The Knot Book. American Mathematical Society. ISBN 0-8050-7380-9.
• Lickorish, W. B. R. (1997). An Introduction to Knot Theory. Graduate Texts in Mathematics. 175. New York: Springer-Verlag. ISBN 0-387-98254-X.
This page was last edited on 2 August 2018, at 13:46
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.