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Studying links via closed braids. V. The unlink


Authors: Joan S. Birman and William W. Menasco
Journal: Trans. Amer. Math. Soc. 329 (1992), 585-606
MSC: Primary 57M25; Secondary 20F36
DOI: https://doi.org/10.1090/S0002-9947-1992-1030509-1
MathSciNet review: 1030509
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Abstract: The main result is a version of Markov's Theorem which does not involve stabilization, in the special case of the $ r$-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most $ 1$. To state the result, we need the concept of an "exchange move", which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theorem $ 1$ shows that exchange moves are the only obstruction to reducing a closed $ n$-braid representative of the $ r$-component unlink to the standard closed $ r$-braid representative, through a sequence of braids of nonincreasing braid index.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1030509-1
Keywords: Knot, link, closed braid, Markov equivalence, stabilization
Article copyright: © Copyright 1992 American Mathematical Society

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