Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [Formula: see text]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [Formula: see text]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

中文翻译：

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由区域交叉变化和弧移移动给出的节点和虚拟节点的 Gordian 复合体

Hirasawa 和 Uchida 将 Gordian 结的复形定义为单纯复形，其顶点是 [公式：见正文] 中的结同位素类。后来 Horiuchi 和 Ohyama 使用 [公式：见文本] 定义了虚拟结的 Gordian 复合体 - 移动和禁止移动。在本文中，我们讨论了基于区域交叉变化的Gordian 复形和基于弧移移动的虚拟结的Gordian 复形。弧移移动是虚拟结图中的局部移动，它导致两个连续交叉点之间的局部方向反转。我们证明了在两个 Gordian 复合体中都存在任意高维单纯形，即通过区域交叉变化和弧位移移动。对于任何给定的结（分别为虚拟结）图，我们构建了一个无限的结族（分别为 虚拟结），这样该家族的任何两个不同成员的距离都会逐个区域交叉变化（分别为弧移移动）。我们证明了构造的虚拟结具有相同的仿射指数多项式。