AI 中文总结
本文推广扭结的指标值概念，引入$\boldsymbol{\textit{B}}$-指标多项式作为扭结不变量，构造特定扭结族并明确其多项式，还研究该多项式的镜像与定向反转行为，确定交叉无cosmetic行为的条件。
AI 中文摘要
扭链是与闭曲面（可非定向）上的链图相关的虚拟链的推广。本文将扭结的指标值概念推广，基于此引入扭结的多项式不变量，称为$\boldsymbol{\textit{B}}$-指标多项式。此外，构造了弧位移数为$n$（$n\boldsymbol{\textit{≥1}}$）的扭结族$\boldsymbol{\textit{\textit{D}_n}}$，并以$n$的形式明确确定其$\boldsymbol{\textit{B}}$-指标多项式。这些结果证明了$\boldsymbol{\textit{B}}$-指标多项式作为扭结不变量的有效性和敏感性。还研究了该多项式在镜像和定向反转下的行为，最后通过研究 cosmetic 交叉变换猜想，确定了交叉不具有 cosmetic 行为的条件。
英文摘要
A twisted link is a generalization of a virtual link associated with link diagrams on closed surfaces, which may be non-orientable. In this paper, we generalize the notion of the index value for twisted knots. Based on this generalization, we introduce a polynomial invariant for twisted knots, called the $\mathcal{B}$-Index polynomial. Furthermore, we construct a family of twisted knots $\{D_n\}_{n\geq 1}$ with arc shift number $n$ and determine their $\mathcal{B}$-Index polynomials explicitly in terms of $n$. These results demonstrate the effectiveness and sensitivity of the $\mathcal{B}$-Index polynomial as an invariant of twisted knots. We also study the behavior of this polynomial under mirror images and orientation reversal. Furthermore, we conclude this paper by investigating the cosmetic crossing change conjecture and establishing a condition under which a crossing does not admit cosmetic behavior.
