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Welded Braids
Fedor Duzhin, NTU
Plan of the talk
1.
2.
3.
4.
5.
6.
Ordinary Artin’s and welded braid groups (geometrical
description)
Artin’s presentation for ordinary and welded braid groups
Group of conjugating automorphisms of the free group
Artin’s representation gives an isomorphism of welded
braids and conjugating automorphisms
Simplicial structure on ordinary and welded braids
Racks and quandles
Ordinary braid groups
A braid is:
n descending strands
joins {(i/(n-1),0,1)} to {(i/(n-1),0,0)}
the strands do not intersect each other
considered up to isotopy in R3
multiplication from top to bottom
the unit braid
1=
•
=
Welded braid groups
A welded braid is:
n descending strands
joins {(i/(n-1),0,1)} to {(i/(n-1),0,0)}
double points (welds) are allowed
some additional moves are allowed:
welds might pass through each other
two consecutive welds cancel
multiplication from top to bottom
the same unit braid
=
=
Presentation for braid groups
The braid group on n strands Bn has the following presentation:
=
=
=
Presentation for welded braids
The welded braid group on n strands Wn has the following presentation:
1
3
2
Presentation for welded braids
The welded braid group on n strands Wn has the following presentation:
3
1
2
6
4
=
=
5
=
Presentation for welded braids
The welded braid group on n strands Wn has the following presentation:
1
2
3
4
5
6
Also, there are some mixed relations:
7
=
Presentation for welded braids
The welded braid group on n strands Wn has the following presentation:
1
4
7
2
5
8
3
6
Also, there are some mixed relations:
=
Double points are allowed to pass
through each other
Presentation for welded braids
The welded braid group on n strands Wn has the following presentation:
1
2
3
4
5
6
7
8
9
Also, there are some mixed relations:
=
Usual isotopy
Presentation for welded braids
Theorem (Roger Fenn, Richárd Rimányi, Colin Rourke)
The welded braid
groupgenerators:
on n strands Wn has the
following presentation:
Transpositions:
Artin’s
Generators:
Relations:
Braid group
relations
Permutation
group relations
Mixed
relations
Presentation for welded braids
Corollary
The welded braid group on n strands Wn has a subgroup isomorphic to the braid
group on n strands Bn and a subgroup isomorphic to the permutation group of n
letters Sn. Together, they generate the whole Wn.
Further, we consider these groups as
Generated by σi
Generated by τi
Free group
Free group Fn:
Generators x0,x1,…,xn-1
No relations
Fn is the fundamental group of
the n-punctured disk
AutFn is the group of
automorphisms of Fn
Mapping class group consists of
isotopy classes of selfhomeomorphisms
x0
x1
xn-1
Artin’s representation
Artin’s representation is obtained from considering braids as mapping classes
The disk is made of rubber
Punctures are holes
The braid is made of wire
The disk is being pushed down along the braid
Theorem The braid group is isomorphic to the
mapping class group of the punctured disk
Artin’s representation
Braids and general automorphisms are
applied to free words on the right
Theorem (Artin)
1. The Artin representation is
faithful
2. The image of the Artin
representation is the set of
automorphisms given by
where
satisfying
Conjugating automorphisms
Definition An automorphism φ:Fn→Fn is called conjugating or of
permutation-conjugacy type if
where
If this permutation μ is identity, then φ:Fn→Fn is called basis-conjugating or
of conjugacy type
Similar to pure braid
Basis-conjugating automorphisms
Theorem (McCool)
The group of basis-conjugating free group automorphisms admits the
following presentation
Generators
Relations
Conjugating automorphisms
Lemma (Savushkina)
The group of conjugating automorphisms admits a presentation with
generators
Conjugating automorphisms
Lemma (Savushkina)
The group of conjugating automorphisms admits a presentation with
generators
relations:
Artin’s representation
The welded braid group Wn does not have an obvious interpretation as
mapping class group as the ordinary braid group does
Nevertheless, Artin’s representation can be easily generalised for it:
Artin’s representation
Theorem (Savushkina)
The Artin representation is an isomorphism of the welded braid group and the
group of conjugating automorphisms.
In other words,
1. Artin’s representation for welded braids is faithful
2. Its image is the set of free group automorphisms given by
where
Artin’s representation
Theorem (Savushkina)
The Artin representation is an isomorphism of the welded braid group and the
group of conjugating automorphisms.
Idea of proof
By direct calculation check that McCool’s generators can be expressed as
formulae in Artin’s generators and permutation group generators
Summary about these groups
Exact
Pure braids
1
Basisconjugating
automorphsis
Braids
Permuations
inclusion
=
Welded braids
=
Conjugating
automorphisms
Permuations
inclusion
Exact
1
Crossed simplicial structure
The braid group is a crossed simplicial group, that is,
Homomorphism to the permutation group
Face-operators
Degeneracy-operators
Simplicial identities
Crossed simplicial relation
Crossed simplicial structure
Face-operators are given by
deleting a strand:
Crossed simplicial structure
Degeneracy-operators are given
by doubling a strand:
Permutative action
The braid group Bn acts on the free group Fn so that
and
commute for any braid a
We call it a permutative action
Crossed simplicial structure
Similarly, the welded braid group is a crossed simplicial group
Homomorphism to the permutation group
Face-operators are also given by deleting a strand
Degeneracy-operators are also given by doubling a strand
Permutative action
The welded braid group Wn acts on the free group Fn so that
and
commute for any welded braid a
Quandles
A quandle is a set with an algebraic operation  such that for any a, b, c the
following statements hold
1. aa=a
2. There is a unique x such that xa=b
3. (ab)c=(ac)(bc)
Given a group G, put  to
be the conjugation.
Then (G,) is a quandle
Theorem (Fenn, Rimányi, Rourke)
The welded braid group on n strands Wn is
isomorphic to the automorphism group
AutFQn of the free quandle of rank n
Racks
A rack is a set with an algebraic operation  such that for any a, b, c the
following statements hold
1. There is a unique x such that xa=b
2. (ab)c=(ac)(bc)
Theorem (Fenn, Rimányi, Rourke)
The automorphism group AutFRn of the free rack of rank n is isomorphic to
the wreath product of the welded braid group Wn with the integers.
Thanks for your attention