Virtual Braids and the L--Move

In this paper we prove a Markov Theorem for virtual braids and for some analogs of this structure. The virtual braid group is the natural companion in the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow the L--move methods to prove the Virtual Markov Theorem. One benefit of this approach is a fully local algebraic formulation of the Theorem.Comment: 42 pages, 42 figures, LaTeX documen

A Categorical Model for the Virtual Braid Group

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang-Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang-Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.Comment: 41 pages, 30 figures, LaTeX documen

Virtual Braids

In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strings) and their isotopies via ribbon surfaces and abstract link diagrams. We also give reduced presentations for the virtual braid group, the flat virtual braid group, the welded braid group and several other categories of braids. The paper includes a discussion of the topological intepretation of the welded braid group in terms of tubes embedded in four-space. A sequel to this paper will give a new proof of a Markov Theorem for virtual braids (and related categories) via the L-move (a technique pioneered for classical braids and braids in three-manifolds by the second author).Comment: 31 pages, 22 figures, LaTeX documen

The meaning of S-D dominance

The dominance of S and D pairs in the description of deformed nuclei is one of the facts that provided sustain to the Interacting Boson Approximation. In Ref.(J. Dukelsky and S. Pittel, Phys. Rev. Lett. 86, 4791, 2001.), using an exactly solvable model with a repulsive pairing interaction between bosons it has been shown that the ground state is described almost completely in terms of S and D bosons. In the present paper we study the excited states obtained within this exactly solvable hamiltonian and show that in order to obtain a rotational spectra all the other degrees of freedom are needed.Comment: Are S and D pairs enough to describe deformed nuclei