The Framed Braid Group and 3-Manifolds

Quantum invariants of framed links from ternary self-distributive cohomology

“…The main purpose of the construction of Theorem 4.3, is to show that the natural invariants associated to the ribbon category coincide with the cocycle invariants defined in Section 3. It is in fact possible to bypass this construction and prove that the twisting morphsism θ α and the braiding morphism c α 2,2 induce a representation of the infinite framed braid group (the inductive limit of the framed braid groups F B n of [18]) similarly to [26], using the linear map Φ b of Section 5 (given below). It is convenient, though, to define a ribbon category from a ternary self-distributive structure and a ternary 2-cocycle since this is suitable for a generalization to multiple compatible self-distributive structures.…”

“…Let (X, T ) be a ternary self-distributive object arising from a set-theoretic ternary quandle Q as above, and [α] ∈ H 2 (Q, A) be fixed. A framed link is represented by the closure of an element b ∈ F B n of the framed braid group on n ribbons [18] where, since twisting of the ribbon and crossings commute, it is assumed that the twists are on top of the braid. Using the same notation of [18],…”

“…A framed link is represented by the closure of an element b ∈ F B n of the framed braid group on n ribbons [18] where, since twisting of the ribbon and crossings commute, it is assumed that the twists are on top of the braid. Using the same notation of [18],…”

Graphical Calculus for the Double Affine Q-Dependent Braid Group

“…The above relations may be familiar to many readers. They appear in the study of framed, or ribbon, braid groups, introduced in [14]. A more in-depth and mathematically rigourous treatment of framed braids or ribbon braids can be found in [15,16], among others.…”

“…necessary. In contrast, for framed braids in an infinitely long strip as in [14] more complicated crossing conditions are needed.…”