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Let ${\mathrm{\text{VB}}}_n$, respectively ${\mathrm{\text{WB}}}_n$ denote the virtual, respectively welded, braid group on $n$-strands. We study their commutator subgroups ${\mathrm{\text{VB}}}_n^{\prime }=[{\mathrm{\text{VB}}}_n,{\mathrm{\text{VB}}}_n]$ and, ${\mathrm{\text{WB}}}_n^{\prime }=[{\mathrm{\text{WB}}}_n,{\mathrm{\text{WB}}}_n]$, respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that ${\mathrm{\text{VB}}}_n^{\prime }$ is finitely generated if and only if $n\ge 4$, and ${\mathrm{\text{WB}}}_n^{\prime }$ is finitely generated for $n\ge 3$. Also, we prove that ${\mathrm{\text{VB}}}_3^{\prime }/{\mathrm{\text{VB}}}_3^{″}={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}^{\infty },{\mathrm{\text{VB}}}_4^{\prime }/{\mathrm{\text{VB}}}_4^{″}={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3,{\mathrm{\text{WB}}}_3^{\prime }/{\mathrm{\text{WB}}}_3^{″}={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus \mathbb{Z},{\mathrm{\text{WB}}}_4^{\prime }/{\mathrm{\text{WB}}}_4^{″}={\mathbb{Z}}_3,$ and for $n\ge 5$ the commutator subgroups ${\mathrm{\text{VB}}}_n^{\prime }$ and ${\mathrm{\text{WB}}}_n^{\prime }$ are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.

This paper studies the biquandle as an invariant of virtual knots and links, analyzing its structure via composition of biquandle morphisms where elementary morphisms are associated with the crossings and virtual crossings in the diagram.

We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.

We define Dynnikov coordinates on virtual braid groups. We prove that they are faithful invariants of virtual 2-braids, and present evidence that they are also very powerful invariants for general virtual braids.

In this paper, we introduce ${\mathbb{Z}}_2$-braids and, more generally, $G$-braids for an arbitrary group $G$. They form a natural group-theoretic counterpart of $G$-knots, see [V. O. Manturov; Reidemeister moves and groups, preprint (2014), arXiv:1412.8691]. The underlying idea used in the construction of these objects — decoration of crossings with some additional information — generalizes an important notion of parity introduced by the second author (see [V. O. Manturov, Parity in knot theory, Sb. Math.201(5) (2010) 693–733]) to different combinatorically geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems.

Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang–Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the PPT algebra, which guides the construction of multipartite symmetric states. The virtual knot theory, having permutation as a virtual crossing, provides a topological language describing quantum computation as having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley–Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang–Baxter equations; and virtual Temperley–Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the ABPK diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies non-trivial unitary braid representations with universal quantum gates, derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.