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 | References | Similar Articles | Additional Information

Abstract: The main result is a version of Markov's Theorem which does not involve stabilization, in the special case of the -component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most . 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 shows that exchange moves are the only obstruction to reducing a closed -braid representative of the -component unlink to the standard closed -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