Abstract
In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.
Highlights
There are interplays between the topological spaces and braid structures with applications in geometric as well as algebraic topology, including knot theory and physical sciences
The braid structures can be classified into two varieties: (1) algebraic structures, namely braid groups, and (2) geometric structures called knots if we consider that ∀i ∈ [1, n], fi(0) = fi(1) preserving the hom( fi([0, 1]), si) property [3]
In view of geometric as well as algebraic topology, the homeomorphic embeddings of two cyclically reverse oriented 2-simplices connected by twisted braid-paths in the covering spaces admit an algebraic variety of a homotopic twisted structure
Summary
There are interplays between the topological spaces and braid structures with applications in geometric as well as algebraic topology, including knot theory and physical sciences. The integrals over the braid-paths have various topological invariant properties [1]. An example of such topological invariance is the countable winding number in lower dimensional topological spaces representing the Gauss linking number. In the case of links forming a Borromean ring, the Gauss linking number vanishes the braid-paths are not separable from the planes within the topological spaces [2]. The braid structures can be classified into two varieties: (1) algebraic structures, namely braid groups, and (2) geometric structures called knots if we consider that ∀i ∈ [1, n], fi(0) = fi(1) preserving the hom( fi([0, 1]), si) property [3]. We briefly present the concept of a homological formal sum because a similar concept is followed in this paper with suitable as well as necessary modifications
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