nLab framed braid group

Context

Group Theory

Manifolds and cobordisms

Knot theory

Contents

Idea

Where an ordinary braid group Br n(Σ)Br_n(\Sigma) is a group of isotopy-classes of plain braids under concatenation, and where the closure of such a braid is a plain link, so a framed braid group (also “ribbon braid group”) consists of isotopy classes of framed braids (ribbon braids closing to framed links), whose strands, in addition to braiding around each other, may also twist in themselves.

Concretely, the plain braid group Br n(Σ)Br_n(\Sigma) has a surjective group homomorphism onto the symmetric group Sym nSym_n and the framed braid group FBr n(Σ)FBr_n(\Sigma) is the wreath product group, relative to this permutation group-structure, with the additive group \mathbb{Z} (“counting the twists” of a strand):

FBr n(Σ)Br n(Σ) nBr n(Σ), FBr_n(\Sigma) \;\; \simeq \;\; \mathbb{Z} \wr Br_n(\Sigma) \;\; \simeq \;\; \mathbb{Z}^n \rtimes Br_n(\Sigma) \,,

hence equivalently the semidirect product group of the direct product group n\mathbb{Z}^n via the group action of Br n(Σ)Sym nBr_n(\Sigma) \to Sym_n given by permuting the \mathbb{Z}-factors.

References

General

As a mapping class group

As the mapping class group of surfaces with framed punctures, and in the context of Reshetikhin-Turaev TQFT:

  • Jens Kristian Egsgaard, Søren Fuglede Jørgensen §1.3 in: The homological content of the Jones representations at q=1q = -1, Journal of Knot Theory and Its Ramifications 25 11 (2016) 1650062 [arXiv:1402.6059, doi:10.1142/S0218216516500620]

  • Marco De Renzi, Azat M. Gainutdinov, Nathan Geer, Bertrand Patureau-Mirand, Ingo Runkel, §3.1 in: Mapping Class Group Representations From Non-Semisimple TQFTs, Commun. Contemp. Math. (2021) 2150091 [arXiv:2010.14852, doi:10.1142/S0219199721500917]

  • Rachel Skipper, Xiaolei Wu, Def. 2.1 in: Homological stability for the ribbon Higman–Thompson groups [arXiv:2106.08751]

  • Iordanis Romaidis, pp 37 in: Mapping class group actions and their applications to 3D gravity, PhD thesis, Hamburg (2022) [ediss:9945]

  • Iordanis Romaidis, Ingo Runkel, p. 8 of: CFT correlators and mapping class group averages [arXiv:2309.14000]

In the context of the Crane-Yetter model:

Last revised on March 17, 2025 at 13:17:17. See the history of this page for a list of all contributions to it.