Conjugacy for positive permutation braids

Hugh R. Morton; Richard J. Hadji

Fundamenta Mathematicae (2005)

  • Volume: 188, Issue: 1, page 155-166
  • ISSN: 0016-2736

Abstract

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Positive permutation braids on n strings, which are defined to be positive n-braids where each pair of strings crosses at most once, form the elementary but non-trivial building blocks in many studies of conjugacy in the braid groups. We consider conjugacy among these elementary braids which close to knots, and show that those which close to the trivial knot or to the trefoil are all conjugate. All such n-braids with the maximum possible crossing number are also shown to be conjugate. We note that conjugacy of these braids for n ≤ 5 depends only on the crossing number. In contrast, we exhibit two such braids on 6 strings with 9 crossings which are not conjugate but whose closures are each isotopic to the (2,5) torus knot.

How to cite

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Hugh R. Morton, and Richard J. Hadji. "Conjugacy for positive permutation braids." Fundamenta Mathematicae 188.1 (2005): 155-166. <http://eudml.org/doc/283123>.

@article{HughR2005,
abstract = { Positive permutation braids on n strings, which are defined to be positive n-braids where each pair of strings crosses at most once, form the elementary but non-trivial building blocks in many studies of conjugacy in the braid groups. We consider conjugacy among these elementary braids which close to knots, and show that those which close to the trivial knot or to the trefoil are all conjugate. All such n-braids with the maximum possible crossing number are also shown to be conjugate. We note that conjugacy of these braids for n ≤ 5 depends only on the crossing number. In contrast, we exhibit two such braids on 6 strings with 9 crossings which are not conjugate but whose closures are each isotopic to the (2,5) torus knot. },
author = {Hugh R. Morton, Richard J. Hadji},
journal = {Fundamenta Mathematicae},
keywords = {Positive permutation braid; conjugacy},
language = {eng},
number = {1},
pages = {155-166},
title = {Conjugacy for positive permutation braids},
url = {http://eudml.org/doc/283123},
volume = {188},
year = {2005},
}

TY - JOUR
AU - Hugh R. Morton
AU - Richard J. Hadji
TI - Conjugacy for positive permutation braids
JO - Fundamenta Mathematicae
PY - 2005
VL - 188
IS - 1
SP - 155
EP - 166
AB - Positive permutation braids on n strings, which are defined to be positive n-braids where each pair of strings crosses at most once, form the elementary but non-trivial building blocks in many studies of conjugacy in the braid groups. We consider conjugacy among these elementary braids which close to knots, and show that those which close to the trivial knot or to the trefoil are all conjugate. All such n-braids with the maximum possible crossing number are also shown to be conjugate. We note that conjugacy of these braids for n ≤ 5 depends only on the crossing number. In contrast, we exhibit two such braids on 6 strings with 9 crossings which are not conjugate but whose closures are each isotopic to the (2,5) torus knot.
LA - eng
KW - Positive permutation braid; conjugacy
UR - http://eudml.org/doc/283123
ER -

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