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Unique Diagram of a Spatial Arc and the Knotting Probability

Received: 2 November 2022     Accepted: 18 November 2022     Published: 29 November 2022
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Abstract

There is a question of how a spatial polygonal arc in 3-space such as a linear molecule, a protein, a non-circular DNA, or a linear polymer, etc. is considered as a knot object. This paper answers it by introducing a notion of knotting probability of a spatial arc. As a tool, the knotting probability of an arc diagram which is invariant under isomorphisms of arc diagrams has been already established by the author, which measures how many non-trivial ribbon surface-knots of genus 2 in the 4-space occur when the arc diagram is regarded as a ribbon chord diagram in a 4D research object. A main task of this paper is to show how to obtain an arc diagram uniquely up to isomorphisms from a given oriented spatial polygonal arc. The image of an oriented spatial polygonal arc under the orthogonal projection from the 3-space to a plane along a unit normal vector is not always any arc diagram. It is shown that an arc diagram unique up to isomorphisms which is determined only by the unit normal vector can be obtained by approximating the projection image of an oriented spatial polygonal arc. By combining the resulting arc diagram with the knotting probability of an arc diagram already established, the knotting probability of every spatial polygonal arc belonging to a unit vector is defined. It is also observed that every oriented spatial polygonal arc (except for the case of a polygonal arc contained in a straight line segment) in 3-space admits a unique orthonormal basis of the 3-space. Thus, the knotting probability of a spatial polygonal arc in 3-space is defined. The knotting probabilities of three concrete examples on oriented spatial polygonal arcs are computed.

Published in Pure and Applied Mathematics Journal (Volume 11, Issue 6)
DOI 10.11648/j.pamj.20221106.12
Page(s) 102-111
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Arc Diagram, Approximation, Spatial Arc, Knotting Probability

References
[1] R. H. Crowell and R. H. Fox, Introduction to knot theory (1963) Ginn and Co.; Re-issue Grad. Texts Math., 57 (1977), Springer Verlag.
[2] T. Deguchi and T. Tsurusaki, A statistical study of random knotting using the Vassiliev invariants, J. Knot Theory Ramifications, 3 (1994), 321-353.
[3] A. Kawauchi, A survey of knot theory, Birkhäuser (1996).
[4] A. Kawauchi, On transforming a spatial graph into a plane graph, Statistical Physics and Topology of Polymers with Ramifications to Structure and Function of DNA and Proteins, Progress of Theoretical Physics Supplement, 191 (2011), 235-244.
[5] A. Kawauchi, A chord diagram of a ribbon surface-link, J. Knot Theory Ramifications, 24 (2015), 1540002 (24pp.).
[6] A. Kawauchi, Knot theory for spatial graphs attached to a surface, Contemporary Mathematics, 670 (2016), 141-169.
[7] A. Kawauchi, Supplement to a chord diagram of a ribbon surface-link, J. Knot Theory Ramifications, 26 (2017) 1750033 (5pp.).
[8] A. Kawauchi, A chord graph constructed from a ribbon surface-link, Contemporary Mathematics, 689 (2017), 125-136. Amer. Math. Soc., Providence, RI, USA.
[9] A. Kawauchi, Faithful equivalence of equivalent ribbon surface-links, J. Knot Theory Ramifications, 27, No. 11 (2018), 1843003 (23 pages).
[10] A. Kawauchi, Knotting probability of an arc diagram, Journal of Knot Theory Ramifications 29 (10) (2020) 2042004 (22 pages).
[11] A. Kawauchi, T. Shibuya and S. Suzuki, Descriptions on surfaces in four-space, I: Normal forms, Math. Sem. Notes, Kobe Univ., 10 (1982), 75-125; II: Singularities and cross-sectional links, Math. Sem. Notes, Kobe Univ. 11 (1983), 31-69.
[12] K. Millett, A. Dobay and A. Stasiak, Linear random knots and their scaling behavior, Macromolecules, 38, (2005) 601-606.
[13] E. Uehara and T. Deguchi, Knotting probability of self-avoiding polygons under a topological constraint, J. Chemical Physics, 147, 094901 (2017).
[14] V. Veblen and J. W. Young, Projective geometry, Ginn and Company, (1910), Boston, Massachusetts, U.S.A.
[15] T. Yanagawa, On ribbon 2-knots I; the 3-manifold bounded by the 2-knot, Osaka J. Math., 6 (1969), 447-464.
Cite This Article
  • APA Style

    Akio Kawauchi. (2022). Unique Diagram of a Spatial Arc and the Knotting Probability. Pure and Applied Mathematics Journal, 11(6), 102-111. https://doi.org/10.11648/j.pamj.20221106.12

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    ACS Style

    Akio Kawauchi. Unique Diagram of a Spatial Arc and the Knotting Probability. Pure Appl. Math. J. 2022, 11(6), 102-111. doi: 10.11648/j.pamj.20221106.12

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    AMA Style

    Akio Kawauchi. Unique Diagram of a Spatial Arc and the Knotting Probability. Pure Appl Math J. 2022;11(6):102-111. doi: 10.11648/j.pamj.20221106.12

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  • @article{10.11648/j.pamj.20221106.12,
      author = {Akio Kawauchi},
      title = {Unique Diagram of a Spatial Arc and the Knotting Probability},
      journal = {Pure and Applied Mathematics Journal},
      volume = {11},
      number = {6},
      pages = {102-111},
      doi = {10.11648/j.pamj.20221106.12},
      url = {https://doi.org/10.11648/j.pamj.20221106.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20221106.12},
      abstract = {There is a question of how a spatial polygonal arc in 3-space such as a linear molecule, a protein, a non-circular DNA, or a linear polymer, etc. is considered as a knot object. This paper answers it by introducing a notion of knotting probability of a spatial arc. As a tool, the knotting probability of an arc diagram which is invariant under isomorphisms of arc diagrams has been already established by the author, which measures how many non-trivial ribbon surface-knots of genus 2 in the 4-space occur when the arc diagram is regarded as a ribbon chord diagram in a 4D research object. A main task of this paper is to show how to obtain an arc diagram uniquely up to isomorphisms from a given oriented spatial polygonal arc. The image of an oriented spatial polygonal arc under the orthogonal projection from the 3-space to a plane along a unit normal vector is not always any arc diagram. It is shown that an arc diagram unique up to isomorphisms which is determined only by the unit normal vector can be obtained by approximating the projection image of an oriented spatial polygonal arc. By combining the resulting arc diagram with the knotting probability of an arc diagram already established, the knotting probability of every spatial polygonal arc belonging to a unit vector is defined. It is also observed that every oriented spatial polygonal arc (except for the case of a polygonal arc contained in a straight line segment) in 3-space admits a unique orthonormal basis of the 3-space. Thus, the knotting probability of a spatial polygonal arc in 3-space is defined. The knotting probabilities of three concrete examples on oriented spatial polygonal arcs are computed.},
     year = {2022}
    }
    

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    AU  - Akio Kawauchi
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    PY  - 2022
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    AB  - There is a question of how a spatial polygonal arc in 3-space such as a linear molecule, a protein, a non-circular DNA, or a linear polymer, etc. is considered as a knot object. This paper answers it by introducing a notion of knotting probability of a spatial arc. As a tool, the knotting probability of an arc diagram which is invariant under isomorphisms of arc diagrams has been already established by the author, which measures how many non-trivial ribbon surface-knots of genus 2 in the 4-space occur when the arc diagram is regarded as a ribbon chord diagram in a 4D research object. A main task of this paper is to show how to obtain an arc diagram uniquely up to isomorphisms from a given oriented spatial polygonal arc. The image of an oriented spatial polygonal arc under the orthogonal projection from the 3-space to a plane along a unit normal vector is not always any arc diagram. It is shown that an arc diagram unique up to isomorphisms which is determined only by the unit normal vector can be obtained by approximating the projection image of an oriented spatial polygonal arc. By combining the resulting arc diagram with the knotting probability of an arc diagram already established, the knotting probability of every spatial polygonal arc belonging to a unit vector is defined. It is also observed that every oriented spatial polygonal arc (except for the case of a polygonal arc contained in a straight line segment) in 3-space admits a unique orthonormal basis of the 3-space. Thus, the knotting probability of a spatial polygonal arc in 3-space is defined. The knotting probabilities of three concrete examples on oriented spatial polygonal arcs are computed.
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Author Information
  • Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, Osaka, Japan

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