一类特殊链环的Kauffman多项式
The Kauffman Polynomials of a Special Class of Links
DOI: 10.12677/AAM.2023.12100407, PDF,   
作者: 徐芷微：辽宁师范大学数学学院，辽宁 大连
关键词: Kauffman多项式；递归关系式；生成函数；Kauffman Polynomial； Recurrence Relation； Generating Function
摘要: Kauffman多项式在纽结理论中占据一定地位，是纽结和链环中最有用的双变量Laurent多项式不变量之一，其已经成为量子拓扑的基本构建块。本文主要研究一类特殊不定向链环——复叠链环，研究了这类链环的Kauffman多项式以及Kauffman多项式对应的生成函数。借助直线型链环的Kauffman多项式对复叠链环的Kauffman多项式进行计算，这为研究定向复叠链环的Kauffman多项式以及BLM/Ho多项式奠定基础。
Abstract: The Kauffman polynomial is probably the most useful two-variable polynomial invariants of knots and links. It generalizes the Jones polynomial, and it has become basic building blocks of quantum topology. In this paper, we mainly study a special type of links—the covering links, and we study the Kauffman polynomials of the link and the corresponding generating functions. The Kauffman poly-nomials of the covering links is calculated by using the Kauffman polynomials of linear links, which lays a foundation for the study of Kauffman polynomials and BLM/Ho polynomials of the oriented covering links.
文章引用：徐芷微. 一类特殊链环的Kauffman多项式[J]. 应用数学进展, 2023, 12(10): 4153-4165. https://doi.org/10.12677/AAM.2023.12100407

参考文献

[1] Berceanu, B. and Nizami, A.R. (2014) A Recurrence Relation for the Jones Polynomial. Journal of the Korean Mathe-matical Society, 51, 443-462. [Google Scholar] [CrossRef] 
[2] Duzhin, S. and Shkolnikov, M. (2015) A Formula for the HOMFLY Polynomial of Rational Links. Arnold Mathematical Journal, 1, 345-359. [Google Scholar] [CrossRef] 
[3] Taşköprü, K. and Altıntaş, İ. (2015) HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials. The Electronic Journal of Combinatorics, 22, P4.8. [Google Scholar] [CrossRef] 
[4] Altıntas, I., Tasköprü, K. and Beyaztas, M. (2018) Bracket Polynomials of To-rus Links as Fibonacci Polynomials. International Journal of Advances in Applied Mathematics and Mechanics, 5, 35-43.
[5] Altintas, I. and Takprü, K. (2019) Unoriented Knot Polynomials of Torus Links as Fibonacci-Type Polyno-mials. Asian-European Journal of Mathematics, 12, No. 4. [Google Scholar] [CrossRef] 
[6] (英)M. A. Armstrong. 基础拓扑学[M]. 北京: 人民邮电出版社, 2010.
[7] 刘卫丽. 关于纽结的一个多项式不变量[D]: [硕士学位论文]. 大连: 大连理工大学, 2014.
[8] Adams, C. (1994) The Knot Book. W.H. Freeman and Company, New York.
[9] Adams C. (2004) The Knot Book. American Mathematical Society, New York.
[10] Rolfsen, D. and Chelsea, A. (1976) Knots and Links. American Mathematical Society, New York.
[11] 颜春惺. 一些定向链环的HOMFLY多项式[D]: [硕士学位论文]. 北京: 中国石油大学, 2021.
[12] Alexander, J. W. (1928) Topological Invariants of Knots and Links. Transactions of the American Mathematical Society, 30, 275-306. [Google Scholar] [CrossRef] 

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