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**Studying links via closed braids. IV: Composite links and split links.**
*(English)*
Zbl 0711.57006

Invent. Math. 102, No. 1, 115-139 (1990); erratum ibid. 160, No. 2, 447-452 (2005).

This is number 4 in a series of papers which study presentation of links as closed braids. The general aim is to find a succession of simple moves between closed braid representatives of the same link, such that the number of strings at each stage is no larger than the maximum for the two braids.

Here the goal is realized in the case of split or composite links, when one of the braid representatives has a simple form as an obvious split or composite braid. The extra move, which is introduced to avoid string- increasing applications of the Markov move, is called here an ‘exchange move’. It leaves the number of strings in the braid unchanged.

The main theorems, proved by the same manipulative techniques of the whole series of papers, are that any closed braid representing a composite or split link can be altered by exchange moves and conjugacies only to be a composite or split link. The major corollary is that (braid index-1) is additive under composition of knots.

Here the goal is realized in the case of split or composite links, when one of the braid representatives has a simple form as an obvious split or composite braid. The extra move, which is introduced to avoid string- increasing applications of the Markov move, is called here an ‘exchange move’. It leaves the number of strings in the braid unchanged.

The main theorems, proved by the same manipulative techniques of the whole series of papers, are that any closed braid representing a composite or split link can be altered by exchange moves and conjugacies only to be a composite or split link. The major corollary is that (braid index-1) is additive under composition of knots.

Reviewer: H.Morton

### MSC:

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

### Keywords:

split links; presentation of links as closed braids; composite links; composite braid; number of strings; exchange moves; braid index
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\textit{J. S. Birman} and \textit{W. W. Menasco}, Invent. Math. 102, No. 1, 115--139 (1990; Zbl 0711.57006)

### References:

[1] | Bennequin, D.: Entrelacements et equations de Pfaff. Asterisque107-108, 87-161 (1983) · Zbl 0573.58022 |

[2] | Birman, J.S.: Braids, links and mapping class groups. Ann. Math. Stud.82 (1974) |

[3] | Birman, J.S., Menasco, W.W.: Studying links via closed braids I: A finiteness theorem. Preprint · Zbl 0724.57001 |

[4] | Birman, J.S., Menasco, W.W.: Studying links via closed braids II: On a theorem of Bennequin, Topology and its Applications (to appear) · Zbl 0722.57001 |

[5] | Birman, J.S., Menasco, W.W.: Studying links via closed braids III: Classifying links which are closed 3-braids. Preprint · Zbl 0813.57010 |

[6] | Birman, J.S., Menasco, W.W.: Studying links via closed braids V: The unlink. Trans. AMS. · Zbl 0758.57005 |

[7] | Garside, F.: The braid groups and other groups. Q. J. Math. Oxford20 (No. 78), 235-254 · Zbl 0194.03303 |

[8] | Markov, A.A.: Uber die freie aquivalenz der geschlossenen Zopfe. Rec. Soc. Math. Moscou43, 73-78 (1936) · Zbl 0014.04202 |

[9] | Morton, H.R.: Closed braids which are not prime knots. Math. Proc. Camb. Philos. Soc.86, 421-426 (1979) · Zbl 0433.57005 |

[10] | Morton, H.R.: Threading knot diagrams. Math. Proc. Camb. Philos. Soc.99, 247-260 (1986) · Zbl 0595.57007 |

[11] | Morton, H.R.: Infinitely many fibered knots with the same Alexander polynomial. Topology17, 101-104 (1978) · Zbl 0383.57005 |

[12] | Murasugi, K.: On closed 3-braids. Mem. AMS151 (1976) · Zbl 0327.55001 |

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