Alexander's theorem

In mathematics Alexander's theorem states that every <a href="/facts/Knot_(mathematics)/UMtKJ36m">knot</a> or <a href="/facts/Link_(knot_theory)/hoFDYmVj">link</a> can be represented as a <a href="/facts/Braid_theory/rn5323Kx">closed</a> <a href="/facts/Braid_theory/rn5323Kx">braid</a>; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after <a href="/facts/James_Waddell_Alexander_II/FMPYbH93">James Waddell Alexander II</a>, who published a proof in 1923.
Braids were first considered as a tool of <a href="/facts/Knot_theory/mb7rtqVP">knot theory</a> 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 in <a href="/facts/Colin_Adams_(mathematician)/0wPaWlWv">Colin Adams</a>'s book.
However, the correspondence between knots and braids is clearly not <a href="/facts/Bijection/ayC7YE3z">one-to-one</a>: 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?
This question is addressed in <a href="/facts/Markov_theorem/v2yPKVzA">Markov's theorem</a>, which gives ‘moves’ relating any two closed braids that represent the same knot.

<ul><li>Sossinsky, Alexei B. (2002). <a href="https://books.google.com/books?id=BewrZeACVKcC&pg=PA17">Knots: Mathematics with a Twist</a>. Cambridge, MA: <a href="/facts/Harvard_University_Press/YluAnq74">Harvard University Press</a>. p. 17. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780674009448. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1941191">1941191</a>.</li></ul>

Alexander's theorem open-in-new

Alexander's theorem