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Braids are algebraic objects described by diagrams; the relation to topology is given by Alexander's theorem which states that every knot or link in three-dimensional Euclidean space is the closure of a braid. The Markov theorem, proved by Russian mathematician Andrei Andreevich Markov Jr. [1] describes the elementary moves generating the ...
Braids, Links, and Mapping Class Groups is a mathematical monograph on braid groups and their applications in low-dimensional topology.It was written by Joan Birman, based on lecture notes by James W. Cannon, [1] and published in 1974 by the Princeton University Press and University of Tokyo Press, as volume 82 of the book series Annals of Mathematics Studies.
In 1935, Andrey Markov Jr. described two moves on braid diagrams that yield equivalence in the corresponding closed braids. [6] A single-move version of Markov's theorem, was published by in 1997. [7] Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class of the closed braid. The ...
This book is considered the first comprehensive treatment of braid theory, introducing the modern theory to the field, and contains the first complete proof of the Markov theorem on braids. [ 8 ] In 1973, she joined the faculty at Barnard College , where she served as Chairman of the Mathematics Department from 1973 to 1987, 1989 to 1991, and ...
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. [1] Braids were first considered as a tool of knot theory by
Knot moves or operations include the flype, Habiro move, Markov moves (I. conjugation and II. stabilization), pass move, Perko move, and Reidemeister moves (I. twist move, II. poke move, and III. slide move).
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In Jones' approach, it resulted from a kind of "trace" of a particular braid representation into an algebra which originally arose while studying certain models, e.g. the Potts model, in statistical mechanics. Let a link L be given. A theorem of Alexander states that it is the trace closure of a braid, say with n strands.