Are these braids equal?

Question

If you are not familiar with Braid-Theory I recommend that you read this first. This question assumes that you are at least familiar with the concepts at hand and assumes you are well familiar with group-theory

Let us define σ_n to be the braid in which the nth strand (One indexed) from the top crosses over the n+1th strand, and σ_n^- to be the inverse of σ_n (That is the n+1th strand crosses the nth strand).

The braid group B_n is then generated by <σ₁,σ₂,σ₃, . . . , σ_n-1>. Thus every braid on B_n can be written as the product of the σ-braids.¹

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Determining if two braids on a group are equal is not a simple task. It may be pretty obvious that σ₁σ₃ = σ₃σ₁, but it is a little less obvious that for example σ₂σ₁σ₂ = σ₁σ₂σ₁.²

So the question is "How can we determine if two braids are the same?". Well the two examples above each represent a bit of this. In general the following relations, called Artin's relations, are true:

• σ_iσ_j = σ_jσ_i; i - j > 1

• σ_iσ_i+1σ_i = σ_i+1σ_iσ_i+1

We can use these two relations in conjunction with the group axioms to prove that any equal braids are equal. Thus two braids are equal iff the repeated application of these relations and the group axioms can demonstrate so.

Task

You will write a program or function to take two braids and determine whether or not they are equal. You may also optionally take a positive integer representing the order of the group.

This is a code-golf question so answers will be scored in bytes, with less bytes being better.

Input and Output

You should represent a Braid as an ordered list of generators, (or any equivalent structure, e.g. vector). You may represent the generators in any reasonable form (e.g. an integer, a two tuple of a positive integer and a boolean).

On par with standard descision-problem rules you should output one of two distinct values, an accept an reject.

Test Cases

[],       []              -> True
[1,-1],   []              -> True
[1,2,1],  [2,1,2]         -> True
[1,3],    [3,1]           -> True
[1,3,2,1],[3,2,1,2]       -> True
[1,4,-4,3,2,1], [3,2,1,2] -> True
[2,2,1],  [2,1,2]         -> False
[1,2,-1], [-1,2,1]        -> False
[1,1,1,2],[1,1,2]         -> False

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^{1: Note that while B_n satisfies all the properties of a group the operation on our braid group is not commutative, and thus our group is not abelian.}

^{2: If you would like to verify this for yourself I suggest applying σ₁- to both sides, If you draw the two out on paper, or model them with actual strings it should become apparent why this is the case.}

Answer 1

Haskell, 190 bytes

i!j|j<0=reverse$map(0-)$i!(-j)|i==j=[i,i+1,-i]|i+1==j=[i]|i+j==0=[j+1]|i+j==1=[-j,-i,j]
_!j=[j]
j%(k:a)|j+k==0=a
j%a=j:a
i&a=foldr(%)[]$foldr((=<<).(!))[i]a
a?n=map(&a)[1..n]
(a#b)n=a?n==b?n

Try it online!

How it works

Let F_n be the free group on n generators x₁, …, x_n. One of the first results in braid theory (Emil Artin, Theorie der Zöpfe, 1925) is that we have an injective homomorphism f : B_n → Aut(F_n) where the action f_σi of σ_i is defined by

f_σi(x_i) = x_ix_{i + 1}x_i⁻¹,
f_σi(x_{i + 1}) = x_i,
f_σi(x_j) = x_j for j ∉ {i, i + 1}.

The inverse f_σi⁻¹ is given by

f_σi⁻¹(x_i) = x_{i + 1},
f_σi⁻¹(x_{i + 1}) = x_{i + 1}⁻¹x_ix_{i + 1},
f_σi⁻¹(x_j) = x_j for j ∉ {i, i + 1}

and of course composition is given by f_ab = f_a ∘ f_b.

To test whether a = b ∈ B_n, it suffices to test that f_a(x_i) = f_b(x_i) for all i = 1, …, n. This is a much simpler problem in F_n, where we only need to know how to cancel x_i with x_i⁻¹.

In the code:

• i!j computes f_σi(x_j) (where either i or j may be negative, representing an inverse),
• foldr(%)[] performs reduction in the free group,
• i&a computes f_a(x_i),
• a?n computes [f_a(x₁), …, f_a(x_n)],
• and (a#b)n is the equality test for a = b in B_n.

Answer 2

Python 2, 270 263 260 250 249 241 bytes

def g(b,i=0):
 while i<len(b)-1:
  R,s=b[i:i+2]
  if R<0<s:b[i:i+2]=[[],[s,-R,-s,R],[s,R]][min(abs(R+s),2)];i=-1
  i+=1
 return b
def f(a,b):
 b=g(a+[-v for v in b][::-1]);i=0
 while i<len(b)and b[0]>0:b=b[1:]+[b[0]];i+=1   
 return g(b)==[]

Try it online!

Implementation of the 'subword reversing' method of solving the braid isotopy problem: a=b iff ab^-1 = the identity.

Algorithm taken from: Efficient solutions to the braid isotopy problem, Patrick Dehornoy; he describes several other algorithms which may be of interest...

This algorithm works by marching left to right in the list, searching for a negative number followed by a positive number; i.e., a sub-word of the form x_i^-1x_j with i,j >0.

It uses the following equivalences:

x_i^-1x_j = x_jx_ix_j^-1x_i^-1 if i=j+1 or j=i+1

x_i^-1x_j= identity (empty list) if i==j

x_i^-1x_j = x_jx_i^-1 otherwise.

By repeated application, we end up with a list of the form w1 + w2, where every element of w1 is positive and every element of w2 is negative. (This is the action of the function g).

We then apply g a second time to the list w2 + w1; the resulting list should be empty iff the original list was equivalent to the identity.