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.

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
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Alexander's theorem

From Wikipedia, the free encyclopedia

This is a typical element of the braid group, which is used in the mathematical field of knot theory.
This is a typical element of the braid group, which is used in the mathematical field of knot theory.

In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid. The theorem is named after James Waddell Alexander II, who published its proof in 1923.

Braids were first considered as a tool of knot theory by Alexander. His theorem gives a positive answer to the question Is it always possible to transform a given knot into a closed braid? A good construction example is found on page 130 of Adams's The Knot book (see ref. below). However, the correspondence between knots and braids is clearly not one-to-one: a knot may have many braid representations. For example, conjugate braids yield equivalent knots. This leads to a second fundamental question: which closed braids represent the same knot type?

That question is addressed in Markov's theorem, which gives ‘moves’ relating any two closed braids that represent the same knot.

YouTube Encyclopedic

  • 1/3
    1 260
    3 913
  • Turaev Surface
  • Braids Solution
  • Constucting the Turaev Surface of Trefoil



  • Alexander, James (1923). "A lemma on a system of knotted curves". Proc. Natl. Acad. Sci. USA. 9: 93–95. Bibcode:1923PNAS....9...93A. doi:10.1073/pnas.9.3.93. PMC 1085274.
  • Sossinsky, A. B. (2002). Knots: Mathematics with a Twist. Harvard University Press. p. 17. ISBN 9780674009448.
  • Adams, Colin C. The Knot Book. AMS. p. 130.

This page was last edited on 2 August 2018, at 08:43
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.