标题： 定向虚拟结的 Reidemeister 移动生成集 显示英文标题

标题： A generating set of Reidemeister moves of oriented virtual knots

摘要： 在有向纽结理论中，验证一个量是否是不变量需要检查它在所有有向Reidemeister移动下的不变性，这个过程可能既复杂又耗时。 一组生成移动简化了这一点，只需验证最小子集中的移动，其他所有移动都可以由此推导出来。 虽然经典有向Reidemeister移动的生成集已经确立，但它们的虚拟对应物研究较少。 在这项研究中，我们枚举了有向虚拟Reidemeister移动，考虑旋转和组合对称性的冗余后，确定了十七个不同的移动。 我们证明了一个四元素子集可以作为这些移动的生成集。 这一结果为验证有向虚拟纽结的不变量提供了一种简便的方法，并为虚拟纽结理论的未来发展奠定了基础，特别是在不变量及其计算属性的研究方面。 显示英文摘要

摘要： In oriented knot theory, verifying a quantity is an invariant involves checking its invariance under all oriented Reidemeister moves, a process that can be intricate and time-consuming. A generating set of oriented moves simplifies this by requiring verification for only a minimal subset from which all other moves can be derived. While generating sets for classical oriented Reidemeister moves are well-established, their virtual counterparts are less explored. In this study, we enumerate the oriented virtual Reidemeister moves, identifying seventeen distinct moves after accounting for redundancies due to rotational and combinatorial symmetries. We prove that a four-element subset serves as a generating set for these moves. This result offers a streamlined approach to verifying invariants of oriented virtual knots and lays the groundwork for future advancements in virtual knot theory, particularly in the study of invariants and their computational properties.