A virtual[Formula: see text]-string is a chord diagram with [Formula: see text] core circles and a collection of arrows between core circles. We consider virtual [Formula: see text]-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams.Given a virtual 1-string [Formula: see text], Turaev associated a based matrix that encodes invariants of the virtual homotopy class of [Formula: see text]. We generalize Turaev’s method to associate a multistring based matrix to a virtual [Formula: see text]-string, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual [Formula: see text]-strings.

中文翻译：

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基于多字符串的矩阵

一个虚拟的[公式：见文本]字符串是一个带有[公式：见文本]核心圆和核心圆之间的箭头集合的和弦图。我们考虑虚拟 [Formula: see text]-strings up to virtual homotopy，flat virtual Reidemeister 在和弦图上移动。给定一个虚拟 1-string [Formula: see text]，Turaev 关联一个编码虚拟不变量的基于矩阵[公式：见正文]的同伦类。我们推广了 Turaev 的方法，将基于多字符串的矩阵与虚拟 [公式：参见文本]-字符串相关联，解决了 Turaev 的一个开放问题，并为虚拟 [公式：参见文本]-字符串的虚拟同伦类构造了类似的不变量。