Studying links via closed braids VI: A non-finiteness theorem

Exchange moves were introduced in an earlier paper by the same authors. They take one closed n-braid representative of a link to another, and can lead to examples where there are infinitely many conjugacy classes of n-braids representing a single link type. THEOREM I. If a link type has infinitely many conjugacy classes of closed n-braid representatives, then n ≥ 4 and the infinitely many classes divide into finitely many equivalence classes under the equivalence relation generated by exchange moves. This theorem is the last of the preliminary steps in the authors’ program for the development of a calculus on links in S³. THEOREM 2. Choose integers n, g ≥ 1. Then there are at most finitely many link types with braid index n and genus g.