Welded Braids

Fedor Duzhin, NTU

Plan of the talk1. Ordinary Artin’s and welded braid groups (geometrical

description)2. Artin’s presentation for ordinary and welded braid groups3. Group of conjugating automorphisms of the free group4. Artin’s representation gives an isomorphism of welded

braids and conjugating automorphisms5. Simplicial structure on ordinary and welded braids6. Racks and quandles

Ordinary braid groupsA 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 groupsA 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 groupsThe braid group on n strands Bn has the following presentation:

= = =

Presentation for welded braidsThe welded braid group on n strands Wn has the following presentation:

1 3

2

Presentation for welded braidsThe welded braid group on n strands Wn has the following presentation:

= = =

1 3 2

4 6 5

Presentation for welded braidsThe welded braid group on n strands Wn has the following presentation:

Also, there are some mixed relations:1 32

4 65

=

7

Presentation for welded braidsThe welded braid group on n strands Wn has the following presentation:

Also, there are some mixed relations:1 324 657

=

Double points are allowed to pass through each other

8

Presentation for welded braidsThe welded braid group on n strands Wn has the following presentation:

Also, there are some mixed relations:1 32

4 65

7 8

=

9

Usual isotopy

Presentation for welded braidsTheorem (Roger Fenn, Richárd Rimányi, Colin Rourke)The welded braid group on n strands Wn has the following presentation:Generators:

Relations: Braid group relations

Permutation group relations

Mixed relations

Artin’s generators: Transpositions:

Presentation for welded braidsCorollaryThe 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 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 self-homeomorphisms

x0 x1 xn-1

Free group

Artin’s representationArtin’s representation is obtained from considering braids as mapping classes

The disk is made of rubberPunctures are holesThe braid is made of wireThe 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 representationBraids and general automorphisms are applied to free words on the right

Theorem (Artin)1. The Artin representation is

faithful2. 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 automorphismsTheorem (McCool)The group of basis-conjugating free group automorphisms admits the following presentation

Generators

Relations

Conjugating automorphismsLemma (Savushkina)The group of conjugating automorphisms admits a presentation withgenerators

Conjugating automorphismsLemma (Savushkina)The group of conjugating automorphisms admits a presentation withgenerators

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 faithful2. 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

Pure braids Braids Permuations

11

Welded braids=

Conjugating automorphisms

Basis-conjugating

automorphsisPermuations

=inclusion

inclusion

Exact

Exact

Crossed simplicial structureThe braid group is a crossed simplicial group, that is,

Homomorphism to the permutation group

Face-operatorsDegeneracy-operators

Simplicial identities Crossed simplicial relation

Crossed simplicial structureFace-operators are given by deleting a strand:

Crossed simplicial structureDegeneracy-operators are given by doubling a strand:

Permutative actionThe braid group Bn acts on the free group Fn so that

commute for any braid a

and

We call it a permutative action

Crossed simplicial structureSimilarly, the welded braid group is a crossed simplicial group

Homomorphism to the permutation group

Face-operators are also given by deleting a strandDegeneracy-operators are also given by doubling a strand

Permutative actionThe welded braid group Wn acts on the free group Fn so that

commute for any welded braid a

and

QuandlesA quandle is a set with an algebraic operation such that for any a, b, c the

following statements hold1. aa=a2. There is a unique x such that xa=b3. (ab)c=(ac)(bc)

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

RacksA rack is a set with an algebraic operation such that for any a, b, c the

following statements hold1. There is a unique x such that xa=b2. (ab)c=(ac)(bc)

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