Theory of Curves and Knots with Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Geometry and Topology".

Deadline for manuscript submissions: closed (31 August 2024) | Viewed by 2425

Special Issue Editors


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Guest Editor
Faculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, 38220 Kosovska Mitrovica, Serbia
Interests: differential geometry of curves and surfaces; geometric knot theory; infinitesimal bending; generalized Riemannian spaces; applications of geometry to natural processes

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Guest Editor
Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, 18 000 Niš, Serbia
Interests: infinitesimal deformations; knots; generalized Riemannian spaces; applied geometry at civil engineering and architecture; computer graphics

Special Issue Information

Dear Colleagues,

We are pleased to announce the Special Issue entitled “Theory of Curves and Knots with Applications” to be published in the journal Axioms.

Curves are the fundamental geometric elements that have a deep history and still represent an area of intense research activity. Theory of curves can be studied from various aspects and serves as a bridge between different mathematical disciplines. Knot theory is concerned with the study of mathematical knots that represent closed curves without self-intersections. The beauty and depth of curves and knots extend beyond pure mathematics, finding applications in areas such as molecular biology and quantum physics.

This Special Issue is devoted to both the foundational concepts of curves and knots as well as its newer, multidisciplinary applications in various fields of study.

The topics covered in this Special Issue include, but are not limited to the following:

  • Various aspects of the theory of curves: differential geometry of curves, topological aspects, algebraic curves, analytic curves, curves on the surfaces, curve deformation, curves in Minkowski space, curves on manifolds, magnetic curves, geodesics, dual curves in dual spaces, fractal curves, etc;
  • Traditional aspects of knot theory: knot invariants, knot polynomials, and braids. Connections with low-dimensional topology, 3-manifolds and higher dimension spaces, algebraic structures, geometry, and combinatorics;
  • Interdisciplinary applications of curves and knots in fields outside of pure mathematics like molecular biology (e.g., DNA knotting and unknotting), physics (e.g., quantum mechanics, statistical mechanics, kinematics), computer science (e.g., algorithms for knot recognition and classification, computer graphics and design), architecture (curved structures and their stability), engineering, etc.

The purpose of this Special Issue is to present the latest research findings in curves and knots, to highlight the diverse applications and to provide a platform for researchers to share knowledge, collaborate, and advance the study and applications of curve and knot theory.

This Special Issue will serve as a contribution to existing literature by bringing together interdisciplinary research that showcases the breadth of applications of curves and knots.

Dr. Marija S. Najdanović
Prof. Dr. Ljubica Velimirovic
Guest Editors

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Keywords

  • geometry of curves
  • curvature
  • curve deformation
  • variational principles
  • dual curves
  • magnetic curves
  • algebraic curves
  • analytic curves
  • geodesics
  • fractal curves
  • curves in Minkowski space
  • curves on manifolds
  • knots
  • knot invariants
  • knot polynomials
  • braid theory
  • topological transformations
  • knot energies
  • link theory
  • DNA knotting
  • knot recognition algorithms
  • quantum knot invariants
  • geometric knot theory
  • virtual knots
  • statistical mechanics
  • kinematics
  • biophysics
  • computer graphics

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Published Papers (3 papers)

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Research

15 pages, 6274 KiB  
Article
On the Torsional Energy of Deformed Curves and Knots
by Svetozar R. Rančić, Ljubica S. Velimirović and Marija S. Najdanović
Axioms 2024, 13(10), 661; https://doi.org/10.3390/axioms13100661 - 25 Sep 2024
Viewed by 365
Abstract
This paper deals with the study of torsional energy (total squared torsion) at infinitesimal bending of curves and knots in three dimensional Euclidean space. During bending, the curve is subject to change, and its properties are changed. The effect that deformation has on [...] Read more.
This paper deals with the study of torsional energy (total squared torsion) at infinitesimal bending of curves and knots in three dimensional Euclidean space. During bending, the curve is subject to change, and its properties are changed. The effect that deformation has on the curve is measured by variations. Here, we observe the infinitesimal bending of the second order and variations of the first and the second order that occur in this occasion. The subjects of study are curves and knots, in particular torus knots. We analyze various examples both analytically and graphically, using our own calculation and visualization software tool. Full article
(This article belongs to the Special Issue Theory of Curves and Knots with Applications)
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16 pages, 1807 KiB  
Article
Quasi-Configurations Derived by Special Arrangements of Lines
by Stefano Innamorati
Axioms 2024, 13(5), 321; https://doi.org/10.3390/axioms13050321 - 11 May 2024
Viewed by 727
Abstract
A quasi-configuration is a point–line incidence structure in which each point is incident with at least three lines and each line is incident with at least three points. We investigate derived quasi-configurations that arise both by duality and intersecting lines of three special [...] Read more.
A quasi-configuration is a point–line incidence structure in which each point is incident with at least three lines and each line is incident with at least three points. We investigate derived quasi-configurations that arise both by duality and intersecting lines of three special arrangements of lines. Sets with few intersection numbers are provided. Full article
(This article belongs to the Special Issue Theory of Curves and Knots with Applications)
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32 pages, 455 KiB  
Article
Spectral Curves for Third-Order ODOs
by Sonia L. Rueda and Maria-Angeles Zurro
Axioms 2024, 13(4), 274; https://doi.org/10.3390/axioms13040274 - 20 Apr 2024
Viewed by 876
Abstract
Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability [...] Read more.
Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem Ly=λy, for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ, defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of Lλ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems. Full article
(This article belongs to the Special Issue Theory of Curves and Knots with Applications)
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