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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.

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Presentation on theme: "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 transcript:

1 Welded Braids Fedor Duzhin, NTU

2 Plan of the talk 1.Ordinary Artin’s and welded braid groups (geometrical description) 2.Artin’s presentation for ordinary and welded braid groups 3.Group of conjugating automorphisms of the free group 4.Artin’s representation gives an isomorphism of welded braids and conjugating automorphisms 5.Simplicial structure on ordinary and welded braids 6.Racks and quandles

3 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 =

4 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 ==

5 Presentation for braid groups The braid group on n strands B n has the following presentation: ===

6 Presentation for welded braids The welded braid group on n strands W n has the following presentation: 13 2

7 Presentation for welded braids The welded braid group on n strands W n has the following presentation: === 1 32 4 65

8 Presentation for welded braids The welded braid group on n strands W n has the following presentation: Also, there are some mixed relations: 132 465 = 7

9 Presentation for welded braids The welded braid group on n strands W n has the following presentation: Also, there are some mixed relations: 132 465 7 = Double points are allowed to pass through each other 8

10 Presentation for welded braids The welded braid group on n strands W n has the following presentation: Also, there are some mixed relations: 132 465 78 = 9 Usual isotopy

11 Presentation for welded braids Theorem (Roger Fenn, Richárd Rimányi, Colin Rourke) The welded braid group on n strands W n has the following presentation: Generators: Relations: Braid group relations Permutation group relations Mixed relations Artin’s generators: Transpositions:

12 Presentation for welded braids Corollary The welded braid group on n strands W n has a subgroup isomorphic to the braid group on n strands B n and a subgroup isomorphic to the permutation group of n letters S n. Together, they generate the whole W n. Further, we consider these groups as Generated by σ i Generated by τ i

13 Free group F n : Generators x 0,x 1,…,x n-1 No relations F n is the fundamental group of the n-punctured disk AutF n is the group of automorphisms of F n Mapping class group consists of isotopy classes of self- homeomorphisms x0x0 x1x1 x n-1 Free group

14 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

15 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

16 Conjugating automorphisms Definition An automorphism φ:F n →F n is called conjugating or of permutation-conjugacy type if where If this permutation μ is identity, then φ:F n →F n is called basis-conjugating or of conjugacy type Similar to pure braid

17 Basis-conjugating automorphisms Theorem (McCool) The group of basis-conjugating free group automorphisms admits the following presentation Generators Relations

18 Conjugating automorphisms Lemma (Savushkina) The group of conjugating automorphisms admits a presentation with generators

19 Conjugating automorphisms Lemma (Savushkina) The group of conjugating automorphisms admits a presentation with generators relations:

20 Artin’s representation The welded braid group W n 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:

21 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

22 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

23 Summary about these groups Pure braidsBraidsPermuations 11 Welded braids = Conjugating automorphisms Basis- conjugating automorphsis Permuations = inclusion Exact

24 Crossed simplicial structure The braid group is a crossed simplicial group, that is, Homomorphism to the permutation group Face-operators Degeneracy-operators Simplicial identitiesCrossed simplicial relation

25 Crossed simplicial structure Face-operators are given by deleting a strand:

26 Crossed simplicial structure Degeneracy-operators are given by doubling a strand:

27 Permutative action The braid group B n acts on the free group F n so that commute for any braid a and We call it a permutative action

28 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

29 Permutative action The welded braid group W n acts on the free group F n so that commute for any welded braid a and

30 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 W n is isomorphic to the automorphism group AutFQ n of the free quandle of rank n

31 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 AutFR n of the free rack of rank n is isomorphic to the wreath product of the welded braid group W n with the integers. Thanks for your attention

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