Studying links via closed braids. V: The unlink. (English) Zbl 0758.57005

[For Parts I, II, IV and VI see Zbl 0724.57001, Zbl 0722.57001, Zbl 0711.57006 and Zbl 0739.57002, respectively.]
The main theorem of this paper is the following: Theorem. Let \(K\) be an arbitrary closed \(n\)-braid representative of the trivial \(r\)-component link \(L\), \(r\geq 1\). Let \(U_ r\) be the standard closed \(r\)-braid representative of \(L\). Then there is a finite sequence of closed braid representatives of \(L\), \(K=K_ 0\to K_ 1\to K_ 2\to \cdots\to K_ m=U_ r\), such that each \(K_{i+1}\) is obtained from \(K_ i\) by one of the following moves: (1) a complexity-reducing exchange move, (2) a complexity-reducing deletion of a trivial loop, (3) a complexity- preserving isotopy in the complement of the axis. From this theorem, we observe that only exchange move that may change conjugacy class of representative is the only obstruction to reducing a closed \(n\)-braid representative of \(L\) to \(U_ r\).


57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
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