AI 中文总结
该研究针对平面虚拟纽托伊德，引入$r$-覆盖确定其同伦关系、推导戈尔迪安距离下界，还研究$\triangle$-移动与虚拟区域交叉变换的对应距离下界，计算了部分同伦对的精确戈尔迪安距离。
AI 中文摘要
本文研究平面虚拟纽托伊德，其同时推广了虚拟纽结与纽托伊德。我们考虑给定的两个平面虚拟纽托伊德能否通过一系列局部移动相互转化，以及最短序列长度的问题。通过引入平面虚拟纽托伊德的$r$-覆盖，确定了不同平面虚拟纽托伊德间的同伦关系，得到它们之间戈尔迪安距离的下界。基于这些结果，我们计算了若干同伦平面虚拟纽托伊德对的精确戈尔迪安距离。此外，我们研究平面虚拟纽托伊德的$\triangle$-移动与虚拟区域交叉变换，通过$r$-覆盖推导了这两种移动对应的戈尔迪安距离下界。
英文摘要
In this paper, we study planar virtual knotoids, which generalize virtual knots and knotoids both. We consider the problem of whether two given planar virtual knotoids can be transformed into each other via a sequence of local moves and what is the shortest sequence length. By introducing $r$-covering of planar virtual knotoids, we determine the homotopy relationship between different planar virtual knotoids, and obtain lower bounds of the Gordian distance between them. On the basis of these results, we demonstrate calculations of the exact Gordian distance of several given pairs of homotopic planar virtual knotoids. Furthermore, we investigate $Δ$-moves and virtual region crossing changes for planar virtual knotoids, and derive lower bounds of the Gordian distance with respect to both moves via $r$-coverings.
Comments30 pages, 27 figures, comments are welcome!
