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Search Results (3)

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Keywords = Dehn twists

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13 pages, 276 KiB  
Article
The Bogomolny–Carioli Twisted Transfer Operators and the Bogomolny–Gauss Mapping Class Group
by Orchidea Maria Lecian
Int. J. Topol. 2025, 2(1), 1; https://doi.org/10.3390/ijt2010001 - 12 Jan 2025
Viewed by 1039
Abstract
The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized PSL (2,Z) group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in [...] Read more.
The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized PSL (2,Z) group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in terms of the Dehn twists. The Bogomolny–Gauss mapping class group of the desymmetrized PSL (2,Z) domain is newly proven. The paradigm to apply the Hecke theory on the CAT spaces on which the Dehn twists act is newly established. The Bogomolny–Gauss map is proven to be one of infinite topological entropy. Full article
12 pages, 949 KiB  
Article
Topological Interactions Between Homotopy and Dehn Twist Varieties
by Susmit Bagchi
Mathematics 2024, 12(20), 3282; https://doi.org/10.3390/math12203282 - 19 Oct 2024
Viewed by 1212
Abstract
The topological Dehn twists have several applications in mathematical sciences as well as in physical sciences. The interplay between homotopy theory and Dehn twists exposes a rich set of properties. This paper generalizes the Dehn twists by proposing the notion of pre-twisted space, [...] Read more.
The topological Dehn twists have several applications in mathematical sciences as well as in physical sciences. The interplay between homotopy theory and Dehn twists exposes a rich set of properties. This paper generalizes the Dehn twists by proposing the notion of pre-twisted space, orientations of twists and the formation of pointed based space under a homeomorphic continuous function. It is shown that the Dehn twisted homotopy under non-retraction admits a left lifting property (LLP) through the local homeomorphism. The LLP extends the principles of Hurewicz fibration by avoiding pullback. Moreover, this paper illustrates that the Dehn twisted homotopy up to a base point in a based space can be formed by considering retraction. As a result, two disjoint continuous functions become point-wise continuous at the base point under retracted homotopy twists. Interestingly, the oriented Dehn twists of a pre-twisted space under homotopy retraction mutually commute in a contractible space. Full article
(This article belongs to the Special Issue Geometry and Topology with Applications)
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30 pages, 13027 KiB  
Article
The Topological Origin of Quantum Randomness
by Stefan Heusler, Paul Schlummer and Malte S. Ubben
Symmetry 2021, 13(4), 581; https://doi.org/10.3390/sym13040581 - 1 Apr 2021
Cited by 6 | Viewed by 3230
Abstract
What is the origin of quantum randomness? Why does the deterministic, unitary time development in Hilbert space (the ‘4π-realm’) lead to a probabilistic behaviour of observables in space-time (the ‘2π-realm’)? We propose a simple topological model for [...] Read more.
What is the origin of quantum randomness? Why does the deterministic, unitary time development in Hilbert space (the ‘4π-realm’) lead to a probabilistic behaviour of observables in space-time (the ‘2π-realm’)? We propose a simple topological model for quantum randomness. Following Kauffmann, we elaborate the mathematical structures that follow from a distinction(A,B) using group theory and topology. Crucially, the 2:1-mapping from SL(2,C) to the Lorentz group SO(3,1) turns out to be responsible for the stochastic nature of observables in quantum physics, as this 2:1-mapping breaks down during interactions. Entanglement leads to a change of topology, such that a distinction between A and B becomes impossible. In this sense, entanglement is the counterpart of a distinction (A,B). While the mathematical formalism involved in our argument based on virtual Dehn twists and torus splitting is non-trivial, the resulting haptic model is so simple that we think it might be suitable for undergraduate courses and maybe even for High school classes. Full article
(This article belongs to the Special Issue Measurements in Quantum Mechanics)
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