4-等价的焊接扩展
A welded extension of 4-equivalence
AI总结：
该研究引入4-等价的焊接扩展，开发适配的箭头演算，实现不含闭分支的代数虚拟缠结有限分类，并证明此类缠结的闭包可简化为 unknot，部分回答了Nakanishi猜想的焊接类似问题。
AI中文摘要：
我们引入一种局部移动，定义了4-等价的焊接扩展。它诱导出与4-等价相同的quandle关系，为经典和焊接两种情形提供了共同的代数框架。我们还开发了一种适配此焊接扩展的箭头演算版本，并将其应用于虚拟缠结的研究。这使得在所得等价关系下，不含闭分支的代数虚拟缠结得到有限分类。作为应用，我们证明了此类缠结的任何闭包都可简化为 unknot，对Nakanishi猜想的焊接类似问题给出了部分肯定回答。
英文摘要：
We introduce a local move defining a welded extension of 4-equivalence. It induces the same quandle relations as 4-equivalence, giving a common algebraic framework for both the classical and welded settings. We also develop a version of arrow calculus adapted to this welded extension and apply it to the study of virtual tangles. This leads to a finite classification of algebraic virtual tangles without closed components under the resulting equivalence relation. As an application, we prove that the closure of any such tangle can be reduced to the unknot, giving a partial positive answer to a welded analogue of Nakanishi's conjecture.
