Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Pure Braid Group

Hannah Lewis

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

DefinitionGeometricAlgebraicConcatenation

Pn

Semi-direct ProductBraid Combing

New PresentationNormal FormWord ProblemConjugacy Problem

Next Step

References

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

At the center of a crypto system is a mathematical trapdoor, thatis, a computational problem that is easy to do in one direction(encryption) but hard to do in reverse (decryption).Mathematicians search for trapdoors that involve computations innon-commutative structures that provide more security in cryptosystems. One such problem is the conjugacy problem in grouptheory. I have been studying the conjugacy problem in the purebraid group.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

There are two ways to look at the braid group Bn

I Geometrically

I Algebraically

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

DefinitionGeometrically, an n-braid is a collection of n disjoint strings wherethe endpoints are fixed.

In Bn the endpoints can be permuted.In Pn the endpoints are not permuted.So Pn is the kernel in the homomorphism g : Bn → Snthat sends a braid to the appropriate permutation of theendpoints. In particular, Pn is a normal subgroup of Bn of index n!.

Figure 1: Braid

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

DefinitionGeometrically, an n-braid is a collection of n disjoint strings wherethe endpoints are fixed.

In Bn the endpoints can be permuted.In Pn the endpoints are not permuted.So Pn is the kernel in the homomorphism g : Bn → Snthat sends a braid to the appropriate permutation of theendpoints. In particular, Pn is a normal subgroup of Bn of index n!.

Figure 1: Braid

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

DefinitionGeometrically, an n-braid is a collection of n disjoint strings wherethe endpoints are fixed.

In Bn the endpoints can be permuted.In Pn the endpoints are not permuted.So Pn is the kernel in the homomorphism g : Bn → Snthat sends a braid to the appropriate permutation of theendpoints. In particular, Pn is a normal subgroup of Bn of index n!.

Figure 1: Braid

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

DefinitionGeometrically an n-braid is a collection of n disjoint strings wherethe endpoints are fixed.

In Bn the endpoints can be permuted.In Pn the endpoints are not permuted.So Pn is the kernel in the homomorphism g : Bn → Snthat sends a braid to the appropriate permutation of theendpoints. In particular, Pn is a normal subgroup of Bn of index n!.

Figure 2: Pure Braid

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Bn has a presentation:〈σ1, ...σn−1 |σiσi+1σi = σi+1σiσi+1, σiσj = σjσi 〉where i = 1, ..., n − 2, j = 1, ..., n − 1, |i − j | > 1

In B3:

In any Bn, σi ’s continue to correspond to the simplest non trivialbraids with one crossing between two adjacent strands.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Bn has a presentation:〈σ1, ...σn−1 |σiσi+1σi = σi+1σiσi+1, σiσj = σjσi 〉where i = 1, ..., n − 2, j = 1, ..., n − 1, |i − j | > 1In B3:

In any Bn, σi ’s continue to correspond to the simplest non trivialbraids with one crossing between two adjacent strands.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Bn has a presentation:〈σ1, ...σn−1 |σiσi+1σi = σi+1σiσi+1, σiσj = σjσi 〉where i = 1, ..., n − 2, j = 1, ..., n − 1, |i − j | > 1In B3:

In any Bn, σi ’s continue to correspond to the simplest non trivialbraids with one crossing between two adjacent strands.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Definition (Multiplication in Pn and Bn)Concatenation of braids. This works both geometrically andalgebraically.

w1 = σ−11 σ−2

3 σ2σ−13

w2 = σ−11 σ3

w1 × w2 = σ−11 σ−2

3 σ2σ−13 σ−1

1 σ3

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

In Pn, if one forgets the nth strand of an n braid, one obtains ann − 1 braid. Thus we have a homomorphism:

f : Pn → Pn−1

The kernel of f , denoted by kerf , turns out to be a free group onn − 1 generators.

In fact we have an isomorphism:

kerf → π1(D − {p1, ..., pn−1}) = F (α1, ..., αn−1)

In P3: this looks like:

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

In Pn, if one forgets the nth strand of an n braid, one obtains ann − 1 braid. Thus we have a homomorphism:

f : Pn → Pn−1

The kernel of f , denoted by kerf , turns out to be a free group onn − 1 generators.In fact we have an isomorphism:

kerf → π1(D − {p1, ..., pn−1}) = F (α1, ..., αn−1)

In P3: this looks like:

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Because we also know that Pn−1 is a subgroup of Pn, the purebraid group on n strands can be written as a semi-direct product:

Pn = F (α1, ..., αn−1) o Pn−1

This can be used to inductively produce presentations of Pn. Forthis we need a presentation of Pn−1, and we need to understandhow Pn−1 acts of the free group F .The first interesting case isn = 3.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

P2 is infinite cyclic generated by z = σ21

This leads to a presentation for P3:

P3 =< α1, α2, z | zα1z−1 = w1, zα2z

−1 = w2 >,

We need to understand how to write w1 and w2 in terms of α1

and α2. It turns out that:w1 = α−1

1 α−12 α1α2α1

w2 = α−11 α2α1

Recall that: α1 = σ22 , α2 = σ2σ

21σ−12

The expressions for w1 and w2 are obtained by combing theappropriate braids.For ease of notation α1 = x , α2 = y .

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

In summary, we obtain the presentation:

〈x , y , z | zxz−1 = x−1y−1xyx , zyz−1 = x−1yx〉

where x = σ22 , y = σ2σ

21σ−12 , z = σ2

1 If we set c = z−1x−1y−1, weobtain the presentation:

< x , y , c |xc = cx , yc = cy >

This shows that P3 is a direct product

F (x , y)× 〈c〉.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Normal FormMove all the c’s to the right using the following relations:xc = cxyc = cy

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Word ProblemAfter putting the word in normal form and free reductions, if theresult is the empty word, then the braid is trivial.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Conjugacy Problem

Suppose we have two words, w1 and w2. We write these words innormal form:

w1 = u1cm1 ,w2 = u2c

m2 ,where ui ∈ F (x , y)

w1 ∼ w2 if and only if m1 = m2 and u1 ∼ u2 in F (x , y)

Recall the conjugacy problem in the free group:Given two words, w1, w2 in F (x , y), cyclically reduce wi to w ′i .Then, w1 ∼ w2 ⇔ w ′2 is a cyclic permutation of w ′1.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Conjugacy Problem

Suppose we have two words, w1 and w2. We write these words innormal form:

w1 = u1cm1 ,w2 = u2c

m2 ,where ui ∈ F (x , y)

w1 ∼ w2 if and only if m1 = m2 and u1 ∼ u2 in F (x , y)

Recall the conjugacy problem in the free group:Given two words, w1, w2 in F (x , y), cyclically reduce wi to w ′i .Then, w1 ∼ w2 ⇔ w ′2 is a cyclic permutation of w ′1.

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

Now that we have generators for P3, we can say:P4 = F (α1, α2, α3) o 〈x , y , c | xc = cx , yc = cy〉Then we start combing the braids

xα1x−1, xα2x

−1, xα3x−1, yα1y

−1, yα2y−1...

Pure Braid Group

Hannah Lewis

Definition

Geometric

Algebraic

Concatenation

Pn

Semi-direct Product

Braid Combing

New Presentation

Normal Form

Word Problem

Conjugacy Problem

Next Step

References

I J. G. Boiser. Computational Problems in the Braid Group.Masters Thesis. San Diego State University. 2009.

I D. Rolfsen. Tutorial on the Braid Group. in Braids:Introductory Lectures on Braids, Configurations and TheirApplications, Lecture Note Series, Institute for MathematicalSciences, National University of Sinapore. Vol 19. WorldScinetific. 2009.