A Knot Invariant Defined Based on the Skein Relation with Two Equations

Liu Weili; Lu Huimin

doi:doi:10.11648/j.acm.20211005.12

| Peer-Reviewed

A Knot Invariant Defined Based on the Skein Relation with Two Equations

Liu Weili, Lu Huimin

Published in Applied and Computational Mathematics (Volume 10, Issue 5)

Received: 9 September 2021 Accepted: 12 October 2021 Published: 16 October 2021

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function f(L), and to prove f(L) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the f_k(L), the property of f(L) is obtained by using the properties of f_k(L). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.

Published in	Applied and Computational Mathematics (Volume 10, Issue 5)
DOI	10.11648/j.acm.20211005.12
Page(s)	114-120
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

Keywords

Knot, Invariants, Reidemeister Moves, Skein Relation

References

[1]	ROLFSEN D. Knots and links [M]. Berkeley: Publish or Perish, 1976.
[2]	ADAMS C C. The Knot Book [J]. American Mathematical Society Providence Ri, 1994: xiv, 307.
[3]	Jiang Leping, Huang Fan Yanli. Figure 8.8.4 Number of chain loop branches of the lattice [J]. Journal of Hunan Institute of Technology: Natural Science Edition, 2012 (4): 17~18.
[4]	Kyeonghui Lee, Young Ho Im, Sera Kim. A family of polynomial invariants for flat virtual knots [J]. Topology and its Applications, 2019, 264.
[5]	Jae Choon Cha, Kent E. Orr, Mark Powell. Whitney towers and abelian invariants of knots [J]. Mathematische Zeitschrift, 2020, 294 (1).
[6]	Min Hoon Kim, Kyungbae Park. An infinite-rank summand of knots with trivial Alexander polynomial [J]. Journal of Symplectic Geometry, 2019, 16 (6).
[7]	Bar Natan Dror, van der Veen Roland. A polynomial time knot polynomial [J]. Proceedings of the American Mathematical Society, 2019, 147 (1).
[8]	Zhiqing Yang. Regional knot invariants [J]. Journal of Knot Theory and Its Ramifications, 2017, 26 (6).
[9]	Manturov Vassily Olegovich. A free-group valued invariant of free knots [J]. Journal of Knot Theory and Its Ramifications, 2021, 30 (06).
[10]	Gabrovšek Boštjan. An invariant for colored bonded knots [J]. Studies in Applied Mathematics, 2021, 146 (3).
[11]	Bardakov Valeriy, Nasybullov Timur. Multi-switches and virtual knot invariants [J]. Topology and its Applications, 2020.
[12]	Joel Hass; Tahl Nowik. Invariants of knot diagrams [J]. Mathematische Annalen. 2008 (1).
[13]	Boyd Rachael, Hepworth Richard, Patzt Peter The homology of the Brauer algebras [J] Selecta Mathematica, 2021, 27 (5).
[14]	Agnese Barbensi, Daniele Celoria. The Reidemeister graph is a complete knot invariant [J]. Algebraic & Geometric Topology, 2020, 20 (2).
[15]	Michael Polyak. Minimal generating sets of Reidemeister moves [J]. Quantum Topology. 2010 (4).
[16]	Vyacheslav Futorny, João Schwarz Algebras of invariant differential operators [J] Journal of Algebra, 2020, 542.
[17]	PRZYTYCKI J H. Survey on recent invariants on classical knot theory [J]. Eprint Arxiv, 2008.

Cite This Article

Plain Text BibTeX RIS

APA Style

Liu Weili, Lu Huimin. (2021). A Knot Invariant Defined Based on the Skein Relation with Two Equations. Applied and Computational Mathematics, 10(5), 114-120. https://doi.org/10.11648/j.acm.20211005.12

Copy | Download

ACS Style

Liu Weili; Lu Huimin. A Knot Invariant Defined Based on the Skein Relation with Two Equations. Appl. Comput. Math. 2021, 10(5), 114-120. doi: 10.11648/j.acm.20211005.12

Copy | Download

AMA Style

Liu Weili, Lu Huimin. A Knot Invariant Defined Based on the Skein Relation with Two Equations. Appl Comput Math. 2021;10(5):114-120. doi: 10.11648/j.acm.20211005.12

Copy | Download

@article{10.11648/j.acm.20211005.12,
  author = {Liu Weili and Lu Huimin},
  title = {A Knot Invariant Defined Based on the Skein Relation with Two Equations},
  journal = {Applied and Computational Mathematics},
  volume = {10},
  number = {5},
  pages = {114-120},
  doi = {10.11648/j.acm.20211005.12},
  url = {https://doi.org/10.11648/j.acm.20211005.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211005.12},
  abstract = {Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function f(L), and to prove f(L) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the fk(L), the property of f(L) is obtained by using the properties of fk(L). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.},
 year = {2021}
}

Copy | Download

TY - JOUR
T1 - A Knot Invariant Defined Based on the Skein Relation with Two Equations
AU - Liu Weili
AU - Lu Huimin
Y1 - 2021/10/16
PY - 2021
N1 - https://doi.org/10.11648/j.acm.20211005.12
DO - 10.11648/j.acm.20211005.12
T2 - Applied and Computational Mathematics
JF - Applied and Computational Mathematics
JO - Applied and Computational Mathematics
SP - 114
EP - 120
PB - Science Publishing Group
SN - 2328-5613
UR - https://doi.org/10.11648/j.acm.20211005.12
AB - Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function f(L), and to prove f(L) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the fk(L), the property of f(L) is obtained by using the properties of fk(L). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.
VL - 10
IS - 5
ER -

Copy | Download

Author Information

Liu Weili

Department of Basic Science, Dalian Naval Academy, Dalian, China
Lu Huimin

Department of Basic Science, Dalian Naval Academy, Dalian, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Liu Weili, Lu Huimin. (2021). A Knot Invariant Defined Based on the Skein Relation with Two Equations. Applied and Computational Mathematics, 10(5), 114-120. https://doi.org/10.11648/j.acm.20211005.12

Copy | Download

ACS Style

Liu Weili; Lu Huimin. A Knot Invariant Defined Based on the Skein Relation with Two Equations. Appl. Comput. Math. 2021, 10(5), 114-120. doi: 10.11648/j.acm.20211005.12

Copy | Download

AMA Style

Liu Weili, Lu Huimin. A Knot Invariant Defined Based on the Skein Relation with Two Equations. Appl Comput Math. 2021;10(5):114-120. doi: 10.11648/j.acm.20211005.12

Copy | Download

@article{10.11648/j.acm.20211005.12,
  author = {Liu Weili and Lu Huimin},
  title = {A Knot Invariant Defined Based on the Skein Relation with Two Equations},
  journal = {Applied and Computational Mathematics},
  volume = {10},
  number = {5},
  pages = {114-120},
  doi = {10.11648/j.acm.20211005.12},
  url = {https://doi.org/10.11648/j.acm.20211005.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211005.12},
  abstract = {Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function f(L), and to prove f(L) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the fk(L), the property of f(L) is obtained by using the properties of fk(L). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.},
 year = {2021}
}

Copy | Download