Studying links via closed braids. V. The unlink

Birman, Joan S.; Menasco, William W.

doi:10.1090/S0002-9947-1992-1030509-1

Studying links via closed braids. V. The unlink
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by Joan S. Birman and William W. Menasco PDF

Trans. Amer. Math. Soc. 329 (1992), 585-606 Request permission

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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