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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. aa=a 2. There is a unique x such that xa=b 3. (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 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 xa=b 2. (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