Inventiones mathematicae

, Volume 102, Issue 1, pp 115–139

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

  • Joan S. Birman
  • William W. Menasco

DOI: 10.1007/BF01233423

Cite this article as:
Birman, J.S. & Menasco, W.W. Invent Math (1990) 102: 115. doi:10.1007/BF01233423


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).

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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