01 January 1970 0 10K Report

Previously, we presented a topological feature of the velocity vector field v(x) as a vector field whose current lines are closed in a circle. On the other hand, we can generalize this definition by assuming that the topological singularities are closed static vector fields of a 7-dimensional sphere S7 whose current lines are narrowed on the product of spheres S3 × S3 × S1. Let us now try to investigate the symmetries of these geometric objects for their compliance with the global gauge symmetries of the so-called standard model of elementary particles. In this regard, it is worth noting that we already have an encouraging result. Indeed, as shown in the previous section, the Lie algebra of linear vector fields that are tangent to the circle S1 lying in the plane is isomorphic to u(1), the Lie algebra of tangent linear vector fields of 3-dimensional sphere S3 lying in 4-dimensional space, which tangent to an arbitrary point of the space S3 × S1, isomorphic to su(2) and the Lie algebra of tangent linear vector fields of 3-dimensional sphere S3, lying in the 6-dimensional space, which tangent to an arbitrary point of the Clifford torus S3 × S3, isomorphic to su(3). In addition, by studying the topology of nodes on the product of spheres S3 × S3 × S1, we can simplify this product to a 3-dimensional torus S1 × S1 × S1 without prejudice to understanding the topological properties of the node, and for clarity consider a closed ribbon lying on a 2-dimensional classical torus S1 × S1. Then, due to the fact that the node on the torus corresponds to the fundamental group of its complement, isomorphic to the corresponding braid group, it is quite fair to compare the elements of this group with the fundamental components of the elementary particle, as is done in the topological Bilson-Thompson model.

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