Alexander's theorem

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.

References

• 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. {{cite journal}}: Invalid |ref=harv (help)
• Sossinsky, A. B. (2002). Knots: Mathematics with a Twist. Harvard University Press. p. 17. ISBN 9780674009448.
• Adams, Colin C. (2004). The Knot Book. Revised reprint of the 1994 original. Providence, RI: American Mathematical Society. p. 130. ISBN 0-8218-3678-1. MR 2079925.