{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:37:00Z","timestamp":1760236620890,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T00:00:00Z","timestamp":1639094400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym13122382","type":"journal-article","created":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T02:07:18Z","timestamp":1639102038000},"page":"2382","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations"],"prefix":"10.3390","volume":"13","author":[{"given":"Susmit","family":"Bagchi","sequence":"first","affiliation":[{"name":"Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660-701, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1023\/A:1010979823190","article-title":"Topological invariants in braid theory","volume":"55","author":"Berger","year":"2001","journal-title":"Lett. 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