Normal forms of random braids
Stephen Tawn (University of Western Sydney)
Monday 17th November, 2014 16:00-17:00 Maths 326
In this talk I will report on joint work with Volker Gebhardt.
Analysing the results of an experiment calculating the normal forms of
a large sample of random braids we observe a rich structure. In
particular we make two conjectures:
1. Except for an initial and a final region whose lengths are
uniformly bounded, the distributions of the factors of the normal form
of sufficiently long random braids depend neither on the position in
the normal form nor on the lengths of the random braids.
2. When multiplying a braid on the right, the expected number of
factors in its normal form that are modified is bounded uniformly in
the length of the braid.
We relate these conjectures to the properties of two regular languages
associated to the normal forms of elements of a Garside monoid. More
specifically, there is a simple condition on the growth rates of these
regular languages that characterises those Garside monoids that
satisfy a variant of the second conjecture. In particular, this
condition holds in all Artin-Tits monoids of spherical type.