Pure Braid Group
Hannah Lewis
Definition
Geometric
Algebraic
Concatenation
Pn
Semi-direct Product
Braid Combing
Pure Braid Group
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
Hannah Lewis
References
Pure Braid Group
Hannah Lewis
Definition
Geometric
Algebraic
Concatenation
Pn
Definition
Geometric
Algebraic
Concatenation
Pn
Semi-direct Product
Braid Combing
New Presentation
Semi-direct Product
Braid Combing
Normal Form
Word Problem
Conjugacy Problem
Next Step
References
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
References
Pure Braid Group
Hannah Lewis
Definition
Geometric
Algebraic
Concatenation
At the center of a crypto system is a mathematical trapdoor, that
is, 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 in
non-commutative structures that provide more security in crypto
systems. One such problem is the conjugacy problem in group
theory. I have been studying the conjugacy problem in the pure
braid group.
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
There are two ways to look at the braid group Bn
I
Geometrically
I
Algebraically
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
References
Definition
Geometrically, an n-braid is a collection of n disjoint strings where
the endpoints are fixed.
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
Geometrically, an n-braid is a collection of n disjoint strings where
the 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 → Sn
that sends a braid to the appropriate permutation of the
endpoints. In particular, Pn is a normal subgroup of Bn of index n!.
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
Geometrically, an n-braid is a collection of n disjoint strings where
the 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 → Sn
that sends a braid to the appropriate permutation of the
endpoints. In particular, Pn is a normal subgroup of Bn of index n!.
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
Figure 1: Braid
Pure Braid Group
Definition
Geometrically an n-braid is a collection of n disjoint strings where
the 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 → Sn
that sends a braid to the appropriate permutation of the
endpoints. In particular, Pn is a normal subgroup of Bn of index n!.
Hannah Lewis
Definition
Geometric
Algebraic
Concatenation
Pn
Semi-direct Product
Braid Combing
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
References
Figure 2: Pure Braid
Pure Braid Group
Hannah Lewis
Definition
Bn has a presentation:
hσ1 , ...σn−1 | σi σi+1 σi = σi+1 σi σi+1 , σi σj = σj σi i
where i = 1, ..., n − 2, j = 1, ..., n − 1, |i − j| > 1
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
Bn has a presentation:
hσ1 , ...σn−1 | σi σi+1 σi = σi+1 σi σi+1 , σi σj = σj σi i
where i = 1, ..., n − 2, j = 1, ..., n − 1, |i − j| > 1
In B3 :
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
Bn has a presentation:
hσ1 , ...σn−1 | σi σi+1 σi = σi+1 σi σi+1 , σi σj = σj σi i
where i = 1, ..., n − 2, j = 1, ..., n − 1, |i − j| > 1
In B3 :
Geometric
Algebraic
Concatenation
Pn
Semi-direct Product
Braid Combing
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
References
In any Bn , σi ’s continue to correspond to the simplest non trivial
braids with one crossing between two adjacent strands.
Pure Braid Group
Definition (Multiplication in Pn and Bn )
Concatenation of braids. This works both geometrically and
algebraically.
Hannah Lewis
Definition
Geometric
Algebraic
Concatenation
Pn
Semi-direct Product
Braid Combing
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
References
w1 = σ1−1 σ3−2 σ2 σ3−1
w2 = σ1−1 σ3
w1 × w2 = σ1−1 σ3−2 σ2 σ3−1 σ1−1 σ3
Pure Braid Group
Hannah Lewis
Definition
In Pn , if one forgets the nth strand of an n braid, one obtains an
n − 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 on
n − 1 generators.
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
In Pn , if one forgets the nth strand of an n braid, one obtains an
n − 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 on
n − 1 generators.
In fact we have an isomorphism:
kerf → π1 (D − {p1 , ..., pn−1 }) = F (α1 , ..., αn−1 )
In P3 : this looks like:
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
Because we also know that Pn−1 is a subgroup of Pn , the pure
braid 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 . For
this we need a presentation of Pn−1 , and we need to understand
how Pn−1 acts of the free group F .The first interesting case is
n = 3.
Pn
Semi-direct Product
Braid Combing
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
References
Pure Braid Group
Hannah Lewis
Definition
P2 is infinite cyclic generated by z = σ12
This leads to a presentation for P3 :
P3 =< α1 , α2 , z | zα1 z −1 = w1 , zα2 z −1 = w2 >,
Geometric
Algebraic
Concatenation
Pn
Semi-direct Product
Braid Combing
New Presentation
We need to understand how to write w1 and w2 in terms of α1
and α2 . It turns out that:
w1 = α1−1 α2−1 α1 α2 α1
w2 =
α1−1 α2 α1
Recall that: α1 = σ22 , α2 = σ2 σ12 σ2−1
The expressions for w1 and w2 are obtained by combing the
appropriate braids.
For ease of notation α1 = x, α2 = y .
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
Pure Braid Group
Hannah Lewis
Definition
Geometric
Algebraic
Concatenation
In summary, we obtain the presentation:
hx, y , z | zxz
−1
=x
−1 −1
y
xyx, zyz
−1
=x
−1
yxi
Pn
Semi-direct Product
Braid Combing
New Presentation
where x = σ22 , y = σ2 σ12 σ2−1 , z = σ12 If we set c = z −1 x −1 y −1 , we
obtain the presentation:
Normal Form
Word Problem
Conjugacy Problem
Next Step
< x, y , c|xc = cx, yc = cy >
This shows that P3 is a direct product
F (x, y ) × hci.
References
Pure Braid Group
Hannah Lewis
Definition
Geometric
Algebraic
Concatenation
Pn
Semi-direct Product
Braid Combing
Normal Form
Move all the c’s to the right using the following relations:
xc = cx
yc = cy
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
Word Problem
After putting the word in normal form and free reductions, if the
result is the empty word, then the braid is trivial.
Normal Form
Word Problem
Conjugacy Problem
Next Step
References
Pure Braid Group
Hannah Lewis
Definition
Conjugacy Problem
Suppose we have two words, w1 and w2 . We write these words in
normal form:
w1 = u1 c m1 , w2 = u2 c m2 ,where ui ∈ F (x, y )
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
Conjugacy Problem
Suppose we have two words, w1 and w2 . We write these words in
normal form:
w1 = u1 c m1 , w2 = u2 c 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 wi0 .
Then, w1 ∼ w2 ⇔ w20 is a cyclic permutation of w10 .
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
Now that we have generators for P3 , we can say:
P4 = F (α1 , α2 , α3 ) o hx, y , c | xc = cx, yc = cy i
Then we start combing the braids
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
xα1 x −1 , xα2 x −1 , xα3 x −1 , y α1 y −1 , y α2 y −1 ...
References
Pure Braid Group
Hannah Lewis
Definition
Geometric
Algebraic
Concatenation
I
I
J. G. Boiser. Computational Problems in the Braid Group.
Masters Thesis. San Diego State University. 2009.
D. Rolfsen. Tutorial on the Braid Group. in Braids:
Introductory Lectures on Braids, Configurations and Their
Applications, Lecture Note Series, Institute for Mathematical
Sciences, National University of Sinapore. Vol 19. World
Scinetific. 2009.
Pn
Semi-direct Product
Braid Combing
New Presentation
Normal Form
Word Problem
Conjugacy Problem
Next Step
References