通过plat表示研究链环：分裂链环与复合链环
Studying links via plats: split and composite links
AI总结：
本文研究如何将分裂或复合链环的任意plat表示转换为可明显识别的表示，证明口袋移动是分裂链环表示的唯一障碍，并利用翻转移动对复合链环得出类似结论。
摘要：
我们的主要结果涉及将分裂或复合链环的任意plat表示转变为一种能被明显识别为分裂或复合的表示。我们利用了在\cite{unlinkviaplats}中首次描述的口袋移动——这种移动在不改变plat表示的链环类型、桥指数或双重陪集的情况下对其进行改变。如果plat表示的平面投影不连通，则该分裂链环的plat表示即为分裂的。我们证明，口袋移动是用分裂plat表示来表示分裂链环的唯一障碍。由于任何口袋移动都对应于一系列双重陪集移动，我们得出推论：分裂链环的每个plat表示的双重陪集都包含一个分裂的plat表示。通过利用翻转移动（同样在第二位作者的论文arXiv:2308.00732 [math.GT]中首次描述），我们针对复合链环获得了类似的结果。
英文摘要：
Our main results concern changing an arbitrary plat presentation of a split or composite link to one which is obviously recognizable as being split or composite. Pocket moves, first described in \cite{unlinkviaplats}, are utilized -- a pocket move alters a plat presentation without changing its link type, its bridge index or the double coset. A plat presentation of a split link is split if the planar projection of the plat presentation is not connected. We prove that pocket moves are the only obstruction to representing split links by split plat presentations. Since any pocket move corresponds to a sequence of double coset moves, we have the corollary that the double coset of every plat presentation of a split link has a split plat presentation. We obtain an analogous result for composite links by utilizing flip moves, which were also first described in the second author's work, arXiv:2308.00732 [math.GT].
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