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Conjugacy for positive permutation braids

Volume 188 / 2005

Hugh R. Morton, Richard J. Hadji Fundamenta Mathematicae 188 (2005), 155-166 MSC: Primary 57M25. DOI: 10.4064/fm188-0-8

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 $n\le 5$ 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.

Authors

  • Hugh R. MortonDepartment of Mathematical Sciences
    University of Liverpool
    Peach St., Liverpool, L69 7ZL, England
    e-mail
  • Richard J. HadjiDepartment of Mathematical Sciences
    University of Liverpool
    Peach St Liverpool, L69 7ZL, England
    e-mail

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