Post #4292318
2026-07-27 08:27 UTC
The formal definition of the Burau representation is a bit daunting, but Vaughan Jones gives a vivid intuitive description. If you think of a braid as a bowling alley with n lanes that can cross over and under each other, and at every crossing there is a probability t of a ball in the upper lane falling down to the lower lane, the (i,j) entry in the Burau matrix is just the probability that a ball starting in lane i exits in lane j.
[This metaphor only works for braids that can be built by stringing together any sequence of the “standard generators” where strand i crosses over strand i+1. To get the whole group you need to throw in the inverses of those generators as well.]
Replies (1)
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@Ianagol@mathstodon.xyz 2026-07-27 15:49
@gregeganSF@mathstodon.xyz Vasudha has claimed that the Gassner representation is faithful, which is maybe a less well-known problem, but probably a bigger deal (it implies linearity of the braid groups with a much simpler to describe representation than the Lawrence-Krammer representation proved to be faithful by Bigelow & Krammer). https://arxiv.org/abs/2208.12378