Pure Virtual Braids Homotopic to the Identity Braid

Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.

💡 Research Summary

The paper investigates the homotopy classification of pure virtual braids, extending the classical notion of braid equivalence by allowing virtual Reidemeister moves, ordinary Reidemeister moves, and self‑crossing changes. After recalling the definition of the pure virtual braid group PVₙ—generated by the usual braid generators σᵢ (real crossings) and virtual generators τᵢ (virtual crossings) subject to the standard braid relations together with commutation relations between σᵢ and τⱼ—the author introduces a homotopy relation that includes “self‑crossing changes,” i.e., the ability to flip the sign of a crossing on a single strand. This operation dramatically enlarges the equivalence classes compared with ordinary isotopy.

The central technical result is a complete description of those elements of PVₙ that are homotopic to the identity braid. By repeatedly applying the mixed relations between σᵢ and τⱼ, any braid β can be rewritten in a normal form consisting of a product of virtual generators and an even power of each real generator. The self‑crossing change then eliminates all odd powers of σᵢ, leaving only a word in the τᵢ’s. Since virtual generators alone can be removed by virtual Reidemeister moves (VR1–VR3), the remaining condition for β to be homotopic to the identity is that its real part be a product of even powers of σᵢ.

For n ≥ 3 the paper identifies a distinguished central element
 z = (σ₁σ₂…σ_{n−1})ⁿ,
which lies in the center of PVₙ. The author proves that z is itself homotopic to the identity, because its even‑power structure can be cancelled by a sequence of self‑crossing changes and virtual moves. Consequently, the entire homotopy class of the identity is generated by the cyclic subgroup ⟨z⟩. In the special case n = 2, the group PV₂ is generated by σ₁ and τ₁, and the analysis shows that any braid reduces to a product of σ₁² and arbitrary τ₁‑words, which again is homotopic to the identity. Thus the homotopy‑trivial pure virtual braids are precisely those lying in the subgroup generated by the even powers of the classical generators together with all virtual generators.

The paper concludes by comparing these findings with earlier work on virtual link homotopy (Kauffman) and virtual braid groups (Bar‑Natan, Kamada). It highlights that the homotopy perspective collapses much of the complexity of PVₙ, revealing a simple algebraic structure governed by a single central element. The author suggests future extensions to non‑pure virtual braids, multi‑strand homotopy, and connections to quantum invariants, indicating that the present classification may serve as a foundation for deeper investigations into virtual knot theory and its algebraic underpinnings.