On representations of Artin–Tits and surface braid groups

Bardakov, Valeriy G.; Bellingeri, Paolo

doi:10.1515/jgt.2010.032

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…, x n of F n , we have that χ n (α)(x j ) = η −1 (x j ), where χ n (α)(x j ) = ρ A (α)(x j ), identifying any usual braid generator σ i with the corresponding α i . It follows that ρ A (α) is an inner automorphism, therefore α belongs to the center of B n (see for instance [3], Remark 1): more precisely α = ((α n−1 • • • α 1 ) n ) m for some m ∈ Z and ρ A (α)(x j ) = (x…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

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The braid group of a necklace

Bellingeri

Bodin

2016

Math. Z.

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We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of (unlinked) circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will call braid group of a necklace) is isomorphic to the braid group over an annulus quotiented by the square of the center. We then define braid groups of necklaces and affine braid groups of type A in terms of automorphisms of free groups and characterize these automorphisms among all automorphisms of free groups. In the case of affine braid groups of type A such a representation is faithful.

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Section: Proof Of Theorem 21mentioning

confidence: 99%

“…It follows that ρ A (α) is an inner automorphism, therefore α belongs to the center of B n (see for instance [3], Remark 1): more precisely α = ((α n−1 . .…”

mentioning

confidence: 99%

The braid group of a necklace

Bellingeri

Bodin

2016

Math. Z.

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“…A straightforward calculation shows that τ 4 (H) = (1, 3), (2,4) , which is isomorphic to S 2 × S 2 . Taking B 2,2 (RP 2 ) to be τ −1 4 ( (1, 3), (2,4) ), the restriction to H of the projection p : B 2,2 (RP 2 ) −→ B 2 (RP 2 ) given geometrically by forgetting the second and fourth strings is an isomorphism. This follows since ker(p) is torsion free and B 2 (RP 2 ) ∼ = Q 16 [35].…”

Section: Appendix On Exact Sequencesmentioning

confidence: 99%

Exact sequences, lower central series and representations of surface braid groups

Bellingeri¹,

Godelle²,

Guaschi³

2011

Preprint

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Homotopy of braids on surfaces: Extending Goldsmith’s answer to Artin

Lima

2019

J. Knot Theory Ramifications

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In 1947, in the paper “Theory of Braids,” Artin raised the question of whether isotopy and homotopy of braids on the disk coincide. Twenty seven years later, Goldsmith answered his question: she proved that in fact the group structures are different, exhibiting a group presentation and showing that the homotopy braid group on the disk is a proper quotient of the Artin braid group on the disk [Formula: see text], denoted by [Formula: see text]. In this paper, we extend Goldsmith’s answer to Artin for closed, connected and orientable surfaces different from the sphere. More specifically, we define the notion of homotopy generalized string links on surfaces, which form a group which is a proper quotient of the braid group on a surface [Formula: see text], denoting it by [Formula: see text]. We then give a presentation of the group [Formula: see text] and find that the Goldsmith presentation is a particular case of our main result, when we consider the surface [Formula: see text] to be the disk. We close with a brief discussion surrounding the importance of having such a fixed construction available in the literature.

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On representations of Artin–Tits and surface braid groups

Cited by 3 publications

References 15 publications

The braid group of a necklace

The braid group of a necklace

Exact sequences, lower central series and representations of surface braid groups

Homotopy of braids on surfaces: Extending Goldsmith’s answer to Artin

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