Inventiones mathematicae

, Volume 102, Issue 1, pp 115–139 | Cite as

Studying links via closed braids IV: composite links and split links

  • Joan S. Birman
  • William W. Menasco
Article

Summary

The main result concerns changing an arbitrary closed braid representative of a split or composite link to one which is obviously recognizable as being split or composite. Exchange moves are introduced; they change the conjugacy class of a closed braid without changing its link type or its braid index. A closed braid representative of a composite (respectively split) link is composite (split) if there is a 2-sphere which realizes the connected sum decomposition (splitting) and meets the braid axis in 2 points. It is proved that exchange moves are the only obstruction to representing composite or split links by composite or split closed braids. A special version of these theorems holds for 3 and 4 braids, answering a question of H. Morton. As an immediate Corollary, it follows that braid index is additive (resp. additive minus 1) under disjoint union (resp. connected sum).

Keywords

Conjugacy Class Disjoint Union Special Version Additive Minus Link Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Joan S. Birman
    • 1
  • William W. Menasco
    • 2
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of MathematicsSUNY at BuffaloBuffaloUSA

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