AI 中文总结
本文提出了区分Q多项式无法区分的无限族扭结的$Q^z$多项式，推导了交叉变化公式等应用，证明其为一阶Vassiliev不变量。
AI 中文摘要
本文介绍了有向扭结的双变量多项式不变量，记为$Q_{K}^{z}(s,t)$，它是N. Kamada与S. Kamada提出的Q多项式的 refinement。我们展示了一个无限族的扭结，它们无法被Q多项式区分，但可被$Q^z$多项式分离。为证明不变性，我们首先确定了有向扭结图的有向Reidemeister移动的生成集，扩展了Ali针对有向虚拟扭结的结果；该结果是新的，具有独立意义，因为它提供了验证任何有向扭结不变量不变性所需的最小框架。作为进一步应用，我们推导了显式的交叉变化公式，得到了同伦扭结间Gordian距离的下界，研究了扭结图中 cosmetic交叉的存在性，最终证明$Q^{z}_{K}(s,t)$是一阶Vassiliev不变量。
英文摘要
This paper introduces a two-variable polynomial invariant for oriented twisted knots, denoted by $Q_{K}^{z}(s,t)$, refining the $Q$-polynomial of N. Kamada and S. Kamada \cite{NaoSei}. We exhibit an infinite family of twisted knots indistinguishable by the $Q$-polynomial but separated by the $Q^z$-polynomial. To prove invariance, we first determine a generating set of oriented Reidemeister moves for twisted knot diagrams, extending the result of Ali \cite{Dan} for oriented virtual knots; this result is new and of independent interest, as it provides the minimal framework needed to verify invariance of any oriented twisted knot invariant. As further applications, we derive an explicit crossing change formula, obtain lower bounds on the Gordian distance between homotopic twisted knots, examine the existence of cosmetic crossings in a twisted knot diagram, and finally prove that $Q^{z}_{K}(s,t)$ is a Vassiliev invariant of order one.
Comments23 pages, 32 figures, 5 tables
