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), 2024. Published by Science Publishing Group |
Arc Diagram, Approximation, Spatial Arc, Knotting Probability
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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
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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Download@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}
}
TY - JOUR T1 - Unique Diagram of a Spatial Arc and the Knotting Probability AU - Akio Kawauchi Y1 - 2022/11/29 PY - 2022 N1 - https://doi.org/10.11648/j.pamj.20221106.12 DO - 10.11648/j.pamj.20221106.12 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 102 EP - 111 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20221106.12 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. VL - 11 IS - 6 ER -
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