Conjugacy for positive permutation braids

Details Conjugacy for positive permutation braids Conjugacy for positive permutation braids

2003-12-10

arxiv.org/abs/math/0312209v1

Fundamenta Mathematicae 188 (2005), 155-166.

Mathematics

Geometric Topology

Scientific paper

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<6 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.

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