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**Studying links via closed braids. VI: A non-finiteness theorem.**
*(English)*
Zbl 0739.57002

Exchange moves take one closed \(n\)-braid representative of a link to another closed \(n\)-braid representative of the link, in general changing the conjugacy class. They can lead to examples where there are infinitely many conjugacy classes of \(n\)-braids representing a single link type. Two conjugacy classes in the braid group \(B_ n\) are “exchange-equivalent” if they have representatives which are equivalent under a finite series of exchange moves.

Theorem: Let \({\mathcal L}\) be a link type which has infinitely many conjugacy classes of closed \(n\)-braid representatives. Then \(n\geq 4\) and the infinitely many conjugacy classes divide into finitely many exchange equivalence classes.

This theorem is the last of the preliminary steps in the authors’ program for the development of a calculus on links in \(S^ 3\).

Theorem: Let \({\mathcal L}\) be a link type which has infinitely many conjugacy classes of closed \(n\)-braid representatives. Then \(n\geq 4\) and the infinitely many conjugacy classes divide into finitely many exchange equivalence classes.

This theorem is the last of the preliminary steps in the authors’ program for the development of a calculus on links in \(S^ 3\).

Reviewer: J.S.Birman

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |