一类链环的Jones多项式

The Jones Polynomials of Class of Knots

摘要: Jones多项式是纽结理论中重要的不变量之一。若两个链环投影图的Jones多项式的不同，则这两个投影图对应着不同的链环，这是研究纽结和链环分类问题的重要方法。Brunnian link是一类既特殊又简单的链环，本文研究一类特殊的Brunnian链B(c₁,c₂,...,c_n)环，本论文给出了B(c₁,c₂,...,c_n)的Jones多项式的计算公式。

Abstract: Knot theory is a branch of mathematical topology and the classification of knots is a very important problem in the field of knots. People solve the problem of knot by looking for knot invariants. Knot invariants are trichrome, crossover number, knot number, bridge number, knot polynomial, knot group, etc. In this paper, we study a class of special Brunnian link. The formula for calculating the Jones polynomial of this kind of chain is given. In order to realize the calculation of B(c₁,c₂,...,c_n) Jones polynomials, on the one hand, by calculating B(c₁) and B(c₂) bracket polynomials, the calculating formulas of B(c₁,c₂,...,c_n) bracket polynomials are obtained. On the other hand, by dividing the given orientation into regions B(c₁,c₂,...,c_n), the enumeration method is used to find the calculation rule of the number of twists, and then the calculation formula of the number of twists B(c₁,c₂,...,c_n) is given. Combining the above two aspects, the calculation formula of the Jones polynomial B(c₁,c₂,...,c_n) is obtained. The innovation of this paper is to generalize the original definition of connected sum, to connect with a specific segment, B(0) to construct B(0,0), then B(0,0) to construct B(0,0,0), and then to find the B(0,0,...0) computational law, which greatly simplifies the computational process.