A Simple and Efficient Algorithm to Identify the Chirality of Polymer Knots Based on the Alexander Polynomial

Qi-Yuan Qiu; Yong-Jian Zhu; Zhong-Tao Wu; Liang Dai

doi:10.1007/s10118-024-3194-5

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A Simple and Efficient Algorithm to Identify the Chirality of Polymer Knots Based on the Alexander Polynomial

RESEARCH ARTICLE | Updated：2024-12-03

- A Simple and Efficient Algorithm to Identify the Chirality of Polymer Knots Based on the Alexander Polynomial
- Chinese Journal of Polymer Science Vol. 42, Issue 12, Pages: 2030-2037(2024)
- Affiliations：
  
  a.Department of Physics, City University of Hong Kong, Hong Kong 999077, China
  b.Shenzhen Research Institute, City University of Hong Kong, Shenzhen 518057, China
  c.Department of Mathematics, The Chinese University of Hong Kong, Hong Kong 999077, China
- Author bio：
  
  liangdai@cityu.edu.hk
- Funds：
- DOI：10.1007/s10118-024-3194-5
  CLC：
- Received：24 May 2024，
  
  Revised：2024-06-12，
  
  Accepted：15 June 2024，
  
  Online First：18 September 2024，
  
  Published：01 December 2024
- Accepted：
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Qiu, Q. Y.; Zhu, Y. J.; Wu, Z. T.; Dai, L. A simple and efficient algorithm to identify the chirality of polymer knots based on the alexander polynomial. Chinese J. Polym. Sci. 2024, 42, 2030–2037

Qi-Yuan Qiu, Yong-Jian Zhu, Zhong-Tao Wu, et al. A Simple and Efficient Algorithm to Identify the Chirality of Polymer Knots Based on the Alexander Polynomial[J]. Chinese Journal of Polymer Science, 2024, 42(12): 2030-2037.
Qiu, Q. Y.; Zhu, Y. J.; Wu, Z. T.; Dai, L. A simple and efficient algorithm to identify the chirality of polymer knots based on the alexander polynomial. Chinese J. Polym. Sci. 2024, 42, 2030–2037 DOI： 10.1007/s10118-024-3194-5.

Qi-Yuan Qiu, Yong-Jian Zhu, Zhong-Tao Wu, et al. A Simple and Efficient Algorithm to Identify the Chirality of Polymer Knots Based on the Alexander Polynomial[J]. Chinese Journal of Polymer Science, 2024, 42(12): 2030-2037. DOI： 10.1007/s10118-024-3194-5.

摘要

We develop a simple and efficient algorithm to identify the chirality of polymer knots

which is very useful in the research of polymer knots. We prove the correctness of this algorithm mathematically.

Abstract

Recent experimental observations of knotting in DNA and proteins have stimulated the simulation studies of polymer knots. Simulation studies usually identify knots in polymer conformations through the calculation of the Alexander polynomial. However

the Alexander polynomial cannot directly discriminate knot chirality

while knot chirality plays important roles in many physical

chemical

and biological properties. In this work

we discover a new relationship for knot chirality and accordingly

develop a new algorithm to extend the applicability of the Alexander polynomial to the identification of knot chirality. Our algorithm adds an extra step in the ordinary calculation of the Alexander polynomial. This extra step only slightly increases the computational cost. The correctness of our algorithm has been proved mathematically by us. The implication of this algorithm in physical research has been demonstrated by our studies of the tube model for polymer knots. Without this algorithm

we would be unable to obtain the tubes for polymer knots.

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references

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