BraidGroup

This small package implements Artin's $B_\infty$ Braid group and some tools.

In particular, it implements:

• Standard group operations, including *, inv, ^ and one
• Artin's free simplification freesimplify!
• Dehornoy reduction reduced, allowing to implement Braid equivalence ==.
• Nice Braid visualizations using Compose
• 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 Braids stores powers of generators.
• Implements both MIME"text/html" and MIME"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.jl is 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:

σ₂⁻²σ₅⁻²σ₄²σ₅⁻¹σ₂⁻²σ₁⁵σ₄σ₂⁻²σ₃⁻⁴σ₂⁻²σ₅σ₁⁻¹σ₃²σ₄²σ₃⁻⁵σ₁⁻²σ₄²σ₁⁻²σ₂⁻²

Note that in notebooks you can choose which MIME type to display. It can be reduced to the unique simplified equivalent braid

julia> reduced(b)