arXiv subjects
Publications and source records attributed to Oleg Chterental.
For $n \geq 2$ we describe an $O(l^3n)$-time algorithm that determines if a length $l$ virtual braid word in the standard presentation of the virtual braid group ${\mathcal VB}_n$ represents the trivial virtual braid.
Yoshikawa [Yo] conjectured that a certain set of moves on marked graph diagrams generates the isotopy relation for surface links in ${\mathbb R}^4$, and this was proved by Swenton [S] and Kearton and Kurlin [KK]. In this paper, we find another proof of this fact for the case of 2-links (surface links with spherical components). The proof involves a version of Roseman's theorem [R] for branch-free broken surface diagrams of 2-links and a construction of marked graphs from branch-free broken surface diagrams.
There is a well known injective homomorphism $ϕ:{\mathcal {B}}_n \rightarrow {\rm Aut}(F_n)$ from the classical braid group ${\mathcal {B}}_n$ into the automorphism group of the free group $F_n$, first described by Artin. This homomorphism induces an action of ${\mathcal {B}}_n$ on $F_n$ that can be recovered by considering the braid group as the mapping class group of $H_n$ (an upper half plane with $n$ punctures) acting naturally on the fundamental group of $H_n$. Kauffman introduced virtual links as an extension of the classical notion of a link in ${\mathbb {R}}^3$. As in the classical case, there is a corresponding group ${\mathcal {VB}}_n$ of virtual braids. In this paper, we will generalize the above action to ${\mathcal {VB}}_n$. We will define a set, ${\mathcal {VCD}}_n$, of "virtual curve diagrams" and define an action of ${\mathcal {VB}}_n$ on ${\mathcal {VCD}}_n$. Then, we will show that, as in Artin's case, the action is faithful. This provides a combinatorial solution to the word problem in ${\mathcal {VB}}_n$. Bardakov and Manturov described an extension $ψ:{\mathcal {VB}}_n\rightarrow {\rm Aut}(F_{n+1})$ of the Artin homomorphism, and raised the question of its injectivity. We find that $ψ$ is not injective by exhibiting a non-trivial virtual braid in the kernel when $n=4$.
We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by physicists. We also introduce quaternionic Hadamard matrices and quaternionic mutually unbiased bases (MUB). In particular we show that the number of MUB in an n-dimensional quaternionic Hilbert space is at most 2n+1. The bound is attained for n=2. We also determine all quaternionic Hadamard matrices of size at most 4.
We examine the SLOCC classification of the (non-normalized) pure states of four qubits obtained by F. Verstraete et al. The rigorous proofs of their basic results are provided and necessary corrections implemented. We use Invariant Theory to solve the problem of equivalence of pure states under SLOCC transformations of determinant 1 and qubit permutations. As a byproduct, we produce a new set of generators for the invariants of the Weyl group of type F_4. We complete the determination of the tensor ranks of 4-qubit pure states initiated by J.-L. Brylinski. As a result we obtain a simple algorithm for computing these ranks. We obtain also a very simple classification of pure states of rank at most 3.