## Studying Links Via Closed Braids V: The Unlink (1992)

Venue: | Trans. Amer. Math. Soc |

Citations: | 8 - 3 self |

### BibTeX

@ARTICLE{Birman92studyinglinks,

author = {Joan S. Birman and William W. Menasco},

title = {Studying Links Via Closed Braids V: The Unlink},

journal = {Trans. Amer. Math. Soc},

year = {1992},

volume = {329},

pages = {585--606}

}

### OpenURL

### Abstract

this paper began with attempts to understand the role of stabilization in Morton's example. Ultimately, we were led to conjecture and prove a special version of Markov's theorem which avoids stabilization, in the case when k is the unlink of r components. To state our results we introduce in Figure 2 the concept of an exchange move. Assume that the braid axis A is the z axis, and that the arc which is labeled n 3 lies in the x-y plane. Up to isotopy of S , an exchange is defined to be an isotopy of K which moves the arc which is labeled n 2 from a position which is a little bit above or below the x-y plane to a position which is a little bit below or above the x-y plane, keeping the rest of K invariant. The labels on the strands mean that a single strand which is labeled "n i " is to be replaced by n i parallel strands. We allow any type of braiding on the n 1 +n 2 (respectively n 1 +n 3 ) strands in the boxes which are labeled X (respectively Y and Z). An exchange move takes nbraids to n-braids, but need not preserve conjugacy class because the isotopy of K in S is not realizable in the complement of the axis A

### Citations

293 |
Hecke algebra representation of braid groups and the link polynomials
- Jones
(Show Context)
Citation Context ... link type are related. Markov's Theorem was announced in [Ma], with a working outline for a proof. It has been important in recent years because of its central role in the theory of knot polynomials =-=[J]-=-. The first complete proof was given in [Bi], using the methods suggested by Markov. Two other proofs have been given more recently in [Be] and [Mo]. THE UNLINK (Version 4.02) 2 Markov's Theorem: Let ... |

225 |
links and the mapping class group
- Birman, Braids
- 1975
(Show Context)
Citation Context ...s announced in [Ma], with a working outline for a proof. It has been important in recent years because of its central role in the theory of knot polynomials [J]. The first complete proof was given in =-=[Bi]-=-, using the methods suggested by Markov. Two other proofs have been given more recently in [Be] and [Mo]. THE UNLINK (Version 4.02) 2 Markov's Theorem: Let K be an arbitrary closed n-braid representat... |

63 |
A lemma on systems of knotted curves
- Alexander
(Show Context)
Citation Context ...equivalent to the assertion that if {H t : ts[0,2p]} is a fibration of S 3 -A by meridian discs then K meets each H t transversally in exactly n coherently oriented intersections. Alexander proved in =-=[A]-=- that every link could be so-represented (in many ways). The theorem which is known as Markov's theorem describes how the various closed braid representatives of a given link type are related. Markov'... |

35 |
Braided surfaces and Seifert ribbons for closed braid
- Rudolph
- 1983
(Show Context)
Citation Context ...connect-summing with a braid representative of the unknot, so that the phenomenon exhibited by his example is pervasive. A similar example for the 2component unlink was given by Rudolph on page 27 of =-=[R]-=-. The work in this paper began with attempts to understand the role of stabilization in Morton's example. Ultimately, we were led to conjecture and prove a special version of Markov's theorem which av... |

10 |
Über die freie Äquivalenz der geschlossenen Zöpfe
- Markov
- 1936
(Show Context)
Citation Context ...o-represented (in many ways). The theorem which is known as Markov's theorem describes how the various closed braid representatives of a given link type are related. Markov's Theorem was announced in =-=[Ma]-=-, with a working outline for a proof. It has been important in recent years because of its central role in the theory of knot polynomials [J]. The first complete proof was given in [Bi], using the met... |

9 | Studying links via closed braids I: a finiteness theorem - Birman, Menasco - 1992 |

9 | An irreducible 4-string braid with unknotted closure - Morton - 1983 |

5 |
Entrelacements et Equations de Pfaff", Asterisque
- Bennequin
- 1983
(Show Context)
Citation Context ...because of its central role in the theory of knot polynomials [J]. The first complete proof was given in [Bi], using the methods suggested by Markov. Two other proofs have been given more recently in =-=[Be]-=- and [Mo]. THE UNLINK (Version 4.02) 2 Markov's Theorem: Let K be an arbitrary closed n-braid representative of k and let K' be an arbitrary closed n'-braid representative. Then there is a finite sequ... |

1 |
l"equivalence libre des tresses fermees
- Weinberg
(Show Context)
Citation Context ...exchange in B n which involves strands of weight 1 can be replaced by a chain in which the intermediate braids have braid index at most n+1. However, that was proved in 1938, by Weinberg, who gave in =-=[W]-=- a very simple replacement sequence. Weinberg's sequence shows that an exchange can be realized by adding one trivial loop, conjugating, and then deleting a trivial loop.|| Comments. The earliest vers... |