In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after James Waddell Alexander II, who published a 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 in Colin Adams's book.
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| - Satz von Alexander (Knotentheorie) (de)
- Alexander's theorem (en)
- アレクサンダーの定理 (ja)
- Teorema de Alexandre (pt)
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| - Der Satz von Alexander ist ein Lehrsatz aus dem mathematischen Gebiet der Knotentheorie. Er besagt, dass jede Verschlingung der Abschluss eines Zopfes ist. Er ermöglicht es, Zopfgruppen für die Untersuchung von Knoten und Verschlingungen nutzbar zu machen. Der Satz von Markov gibt hinreichende und notwendige Bedingungen, wann die Abschlüsse zweier Zöpfe äquivalente Verschlingungen ergeben. Er ist nach dem US-amerikanischen Mathematiker James Alexander (1888–1971) benannt. (de)
- 数学において、アレクサンダーの定理(Alexander's theorem)は、すべての結び目、あるいは絡み目は閉じたブレイドとして表現することができるという定理である。定理の命名は、(J. W. Alexander)に因んでいる。 (closed braid)は、最初はアレクサンダーにより結び目理論のツールとして考え出された。このことから結び目とブレイドに関する 2つの次のような基本的な問題を直接、定式化することができる。第一に、 与えられた結び目を常に閉ブレイドへ変換することが可能か否か？ アレクサンダーの定理 は、この問題への肯定的な答えを与える。結び目とブレイドの間の対応が１対１でないことは明らかであり（たとえば、共役ブレイドは同値な結び目をもたらす）、このことから第二の問題が自然に導かれる。 どのような閉ブレイドが、同一な形の結び目を表現するのか？ この問題へ答えるのが、マルコフの定理であり、任意の 2つのブレイドを関係つける「移動」(move)を与える。 (ja)
- In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after James Waddell Alexander II, who published a 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 in Colin Adams's book. (en)
- Em matemática, o teorema de Alexandre afirma que cada nó ou ligação pode ser representada como um sistema fechado de trança. O teorema é nomeado em honra de James Waddell Alexandre II, que publicou a sua prova, em 1923. Essa questão é abordada no teorema de Markov, o que dá 'movimentos' relacionando quaisquer duas tranças fechadas que representam o mesmo nó. (pt)
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| - In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after James Waddell Alexander II, who published a 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 in Colin Adams's book. 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?This question is addressed in Markov's theorem, which gives ‘moves’ relating any two closed braids that represent the same knot. (en)
- Der Satz von Alexander ist ein Lehrsatz aus dem mathematischen Gebiet der Knotentheorie. Er besagt, dass jede Verschlingung der Abschluss eines Zopfes ist. Er ermöglicht es, Zopfgruppen für die Untersuchung von Knoten und Verschlingungen nutzbar zu machen. Der Satz von Markov gibt hinreichende und notwendige Bedingungen, wann die Abschlüsse zweier Zöpfe äquivalente Verschlingungen ergeben. Er ist nach dem US-amerikanischen Mathematiker James Alexander (1888–1971) benannt. (de)
- 数学において、アレクサンダーの定理(Alexander's theorem)は、すべての結び目、あるいは絡み目は閉じたブレイドとして表現することができるという定理である。定理の命名は、(J. W. Alexander)に因んでいる。 (closed braid)は、最初はアレクサンダーにより結び目理論のツールとして考え出された。このことから結び目とブレイドに関する 2つの次のような基本的な問題を直接、定式化することができる。第一に、 与えられた結び目を常に閉ブレイドへ変換することが可能か否か？ アレクサンダーの定理 は、この問題への肯定的な答えを与える。結び目とブレイドの間の対応が１対１でないことは明らかであり（たとえば、共役ブレイドは同値な結び目をもたらす）、このことから第二の問題が自然に導かれる。 どのような閉ブレイドが、同一な形の結び目を表現するのか？ この問題へ答えるのが、マルコフの定理であり、任意の 2つのブレイドを関係つける「移動」(move)を与える。 (ja)
- Em matemática, o teorema de Alexandre afirma que cada nó ou ligação pode ser representada como um sistema fechado de trança. O teorema é nomeado em honra de James Waddell Alexandre II, que publicou a sua prova, em 1923. Tranças foram inicialmente considerados como uma ferramenta da teoria do nó de Alexander. Seu teorema dá uma resposta positiva para a pergunta: É sempre possível transformar um determinado nó em um sistema fechado de trança? Um bom exemplo de construção encontra-se na página 130 do livro, "O Livro do Nó", de Adams (veja a ref. abaixo).No entanto, a correspondência entre nós e tranças é, claramente, de um-para-um: um nó pode ter muitas representações de trança. Por exemplo, tranças conjugadas produzem nós equivalente. Isto leva a uma segunda questão fundamental: quais tranças fechadas representam o mesmo tipo de nó? Essa questão é abordada no teorema de Markov, o que dá 'movimentos' relacionando quaisquer duas tranças fechadas que representam o mesmo nó. (pt)
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