This small package implements Artin's
In particular, it implements:
- Standard group operations, including
*,inv,^andone - Artin's free simplification
freesimplify! - Dehornoy reduction
reduced, allowing to implement Braid equivalence==. - Nice
Braidvisualizations usingCompose - Braid minimization (Alg 2.2) in Kotov et al
- Compressed braid visualization, with parallel strand crossings. This is not the default, invoque with
composed(braid, compressed=true). This has been used to produce the braid above.
It is similar in scope to Braids.jl from which it draws some inspiration. At difference with it, this package:
- Represents the
$B_\infty$ group rather than$B_n$ - Internally represent braids as product of generators, whereas
Braidsstores powers of generators. - Implements both
MIME"text/html"andMIME"text/plain"output - Operations are done in-place as much as possible: in particular,
reduced!is almost non-allocating, and each H iteration is linear in the length of the braid. - On the negative side, the interesting matrix representation included in
Braids.jlis not implemented.
julia> b = prod(braid(rand(1:5))^rand(-2:2) for i=1:40)displays in MIME"text/html" (e.g. in a notebook) as:
and in plain text as:
σ₂⁻²σ₅⁻²σ₄²σ₅⁻¹σ₂⁻²σ₁⁵σ₄σ₂⁻²σ₃⁻⁴σ₂⁻²σ₅σ₁⁻¹σ₃²σ₄²σ₃⁻⁵σ₁⁻²σ₄²σ₁⁻²σ₂⁻²Note that in notebooks you can choose which MIME type to display.
It can be reduced to the unique simplified equivalent braid
julia> reduced(b)