Representations of Gn3 and related groups

In this paper, we study representations of [Formula: see text]-like groups. The group [Formula: see text] itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of [Formula: see text]-like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on the [Formula: see text]-like group structure.

Quantum genetic spiral the space-time

The nature of quantum mechanics has various interpretations. In this paper we consider the hypothesis of quantum Darwinism. Quantum theory is closely connected with the concept of information. Perhaps there is an analogue of the genetic code for quantum Darwinism. Here the attempt of the genetic formulation of quantum gravity. It is based on the idea of the quantum genetic spiral the space-time, directed along the time axis. Such a mathematical form exists in braid theory. Matter how information is coded in the genetic structure of space-time. Natural and artificial selection of quantum Darwinism leads to the collapse of the wave function and the identification of a dominant gene.

Genetic quantum gravity and quantum Darwinism

The nature of quantum mechanics has various interpretations. In this paper we consider the hypothesis of quantum Darwinism. Quantum theory is closely connected with the concept of information. Perhaps there is an analogue of the genetic code for quantum Darwinism. Here the attempt of the genetic formulation of quantum gravity. It is based on the idea of the quantum of the DNA helix in space-time , directed along the time axis. Twisting together all of the genetic spirals creates the very fabric of space-time. Such a mathematical form exists in braid theory. Matter how information is encoded in the genetic DNA structure of space-time. Natural and artificial selection of quantum Darwinism leads to the collapse of the wave function and the identification of a dominant gene.

Closed braids and knot holders associated to some laser dynamical systems: A pump-modulated Nd-doped Â…ber laser

In this work the arising knots and links for the pump-modulated Nd-doped fiber laser is investigated. For the associated templates, some of their topological invariants, such as braid linking matrix, braid words, crossing number and linking number, are studied. It is recognized that the derived topological invariants are quietly dependent on the control parameters Using the tools of the braid theory, it is shown that pump-modulated Nd-doped fiber laser knots and links are positive fibered knots and links.

On marked braid groups

In this paper, we introduce [Formula: see text]-braids and, more generally, [Formula: see text]-braids for an arbitrary group [Formula: see text]. They form a natural group-theoretic counterpart of [Formula: see text]-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.

Rotations of 4nπ Research and no Twisted Mechanism Example Analysis Based on the Theory of Braid Group

The Cable or Pipe between the Two Relatively Rotating Platforms Exists the Twisted Problem. from Viewpoint of Braid Group Theory, Rope’s Twisted State is Researched on. Based on the Characteristics of a Special Garside Braids Δn and Δnk , the Equivalence of Two Rotation Modes is Revealed. also, that the Determination of Rotation Mode’s Minimum Rotate Range is 4∏ while Using Braid Theory is Proposed. the Theory of Braid Group can Not only be Used as a Criterion for Determine Whether the Cable is Twisted, Finally, but also can be Used as Avoid Cable Twisted during Mechanism Design.