Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister moves, one sometimes needs to insert excess crossings, there are significant gaps between the best known lower and upper bounds on the required number of these added crossings. In this article, we study the problem of turning a diagram of a split link into a split diagram, and we show that there exist split links with diagrams requiring an arbitrarily large number of such additional crossings. More precisely, we provide a family of diagrams of split links, so that any sequence of Reidemeister moves transforming a diagram with $c$ crossings into a split diagram requires going through a diagram with $\Omega(\sqrt{c})$ extra crossings. Our proof relies on the framework of bubble tangles, as introduced by Lunel and de Mesmay, and a technique of Chambers and Liokumovitch to turn homotopies into isotopies in the context of Riemannian geometry.

翻译：在环境空间中，结与链环的形变可以通过称为Reidemeister移动的局部修改在其图式上进行组合研究。尽管众所周知，为了通过Reidemeister移动在等价图式之间转换，有时需要插入额外的交叉点，但关于所需额外交叉点数量的已知最佳下界与上界之间仍存在显著差距。本文研究将分裂链环的图式转化为分裂图式的问题，并证明存在一类分裂链环的图式需要任意大量此类额外交叉点。更精确地说，我们构造了一个分裂链环的图式族，使得任何将具有 $c$ 个交叉点的图式转化为分裂图式的Reidemeister移动序列，都必须经过具有 $\Omega(\sqrt{c})$ 个额外交叉点的中间图式。我们的证明依赖于Lunel和de Mesmay引入的泡状缠结框架，以及Chambers和Liokumovitch在黎曼几何背景下将同伦转化为等距的技术。