nLab framed braid group

Idea

Where an ordinary braid group $Br_n(\Sigma)$ 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 $Br_n(\Sigma)$ has a surjective group homomorphism onto the symmetric group $Sym_n$ and the framed braid group $FBr_n(\Sigma)$ is the wreath product group, relative to this permutation group-structure, with the additive group $\mathbb{Z}$ (“counting the twists” of a strand):

$$FBr_n(\Sigma) \;\; \simeq \;\; \mathbb{Z} \wr Br_n(\Sigma) \;\; \simeq \;\; \mathbb{Z}^n \rtimes Br_n(\Sigma) \,,$$

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

References

General

• Ki Hyoung Ko, Lawrence Smolinsky: The framed braid group and 3-manifolds, Proc. Amer. Math. Soc. 115 (1992) 541-551 [doi:10.1090/S0002-9939-1992-1126197-1, pdf]

• Ki Hyoung Ko, Lawrence Smolinsky: The framed braid group and representations, in: Knots 90, Proceedings in Mathematics, De Gruyter (1992) 289-297 [doi:10.1515/9783110875911, pdf]

• Hans Wenzel, p. 4 of: Braids and invariants of 3-manifolds, Invent Math 114 (1993) 235–275 [doi:10.1007/BF01232670, eudml:144148]

• R. Krasauskas: Crossed simplicial groups of framed braids and mapping class groups of surfaces, Lith Math J 36 (1996) 263–281 [doi:10.1007/BF02986853]

• Nathalie Wahl, p. 25 of: Ribbon braids and related operads, PhD thesis, University of Oxford (2001) [pdf]

• Ralph Cohen, Alexander Voronov: Notes on string topology, in: String topology and cyclic homology, Advanced courses in mathematics CRM Barcelona, Birkhäuser (2006) [math.GT/05036259, doi:10.1007/3-7643-7388-1, pdf]

  (in the context of string topology)

• Ralph Cohen (notes by Eric Malm), p 20 of: MATH 283: Topological Field Theories (2008) [pdf, pdf]

• Paolo Bellingeri, Sylvain Gervais: Surface framed braids, Geom Dedicata 159 (2012) 51–69 [doi:10.1007/s10711-011-9645-5, arXiv:1001.4471]

• Akishi Ikeda: Homological and Monodromy Representations of Framed Braid Groups, Commun. Math. Phys. 359 (2018) 1091–1121 [doi:10.1007/s00220-017-3036-1, arXiv:1702.03918]

• Lukas Woike: The Cyclic and Modular Microcosm Principle in Quantum Topology [arXiv:2408.02644]

  (in relation to the framed little disk operad)

• Anastasios Kokkinakis: Framed Braid Equivalences [arXiv:2503.05342]

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 $q = -1$, 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:

• Ying Hong Tham: §3.2.1 in: On the Category of Boundary Values in the Extended Crane-Yetter TQFT, PhD thesis, Stony Brook (2021) [arXiv:2108.13467]