Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups...

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Welded Braids Fedor Duzhin, NTU

description

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 R 3 multiplication from top to bottom the unit braid =1 =

Transcript of Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups...

Page 1: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

Welded Braids

Fedor Duzhin, NTU

Page 2: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 3: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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 =

Page 4: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

= =

Page 5: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

Presentation for braid groupsThe braid group on n strands Bn has the following presentation:

= = =

Page 6: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

1 3

2

Page 7: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

= = =

1 3 2

4 6 5

Page 8: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 9: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 10: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 11: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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:

Page 12: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 13: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 14: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 15: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 16: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 17: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Generators

Relations

Page 18: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 19: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

relations:

Page 20: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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:

Page 21: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 22: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 23: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

Summary about these groups

Pure braids Braids Permuations

11

Welded braids=

Conjugating automorphisms

Basis-conjugating

automorphsisPermuations

=inclusion

inclusion

Exact

Exact

Page 24: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 25: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 26: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 27: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 28: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 29: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

commute for any welded braid a

and

Page 30: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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

Page 31: Welded Braids Fedor Duzhin, NTU. Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary.

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


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Welded Connections

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ARTIN’S AXIOMS Contents

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