Alexander's theorem

In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid, named after J. W. Alexander.

A closed braid was firstly considered as a tool of knot theory by J. W. Alexander. This leads us directly to formulate two fundamental questions about knots and braids. First,

is it always possible to transform a given knot into a closed braid?

Alexander's theorem answers affirmatively to this question in Alexander (1923). The correspondence between knots and braids is clearly not one-to-one (for example, conjugate braids yield equivalent knots), leading naturally to the second question:

which closed braids represent the same knot type?

That question is addressed in Markov's theorem, which gives ‘moves’ relating any two closed braid.

references

• Alexander, James (1923). "A lemma on a system of knotted curves". Proc. Nat. Acad. Sci. USA. 9: 93–95. {{cite journal}}: Invalid |ref=harv (help)