|
|
一类特殊链环的Kauffman多项式
|
Abstract:
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。
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.
| [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. https://doi.org/10.4134/JKMS.2014.51.3.443 |
| [2] | Duzhin, S. and Shkolnikov, M. (2015) A Formula for the HOMFLY Polynomial of Rational Links. Arnold Mathematical Journal, 1, 345-359. https://doi.org/10.1007/s40598-015-0013-7 |
| [3] | Ta？k？prü, K. and Alt？nta？, ？. (2015) HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials. The Electronic Journal of Combinatorics, 22, P4.8. https://doi.org/10.37236/5324 |
| [4] | Alt？ntas, I., Task？prü, 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. https://doi.org/10.1142/S1793557119500530 |
| [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. https://doi.org/10.1090/S0002-9947-1928-1501429-1 |
