AI 中文总结
本文通过穷举计算机搜索证明，平凡结的花瓣图在最多7个花瓣时可单调简化，但9个花瓣时存在108个困难图，其中三个在对称性下本质不同，揭示了花瓣图与矩形图在简化性质上的差异。
AI 中文摘要
一个结的花瓣图是一个具有单个多交叉且没有嵌套环的投影；它由穿过该多交叉的弦线高度的排列来编码。Colton、Glover、Hughes 和 Sandberg 证明了花瓣图的 Reidemeister 型定理：两个花瓣排列表示同一个结，当且仅当它们可以通过平凡的花瓣添加和删除以及交叉交换相互关联。我们提出一个问题：是否每个平凡结的花瓣图都可以在不增加花瓣数的情况下简化为单花瓣图，这类似于 Dynnikov 关于矩形图的单调简化定理。通过穷举且经过认证的计算机搜索，我们证明这对于最多 7 个花瓣的图成立，而对于 9 个花瓣的图则不成立。在 40320 个具有 9 个花瓣的花瓣图中，24992 个表示平凡结，其中恰好有 108 个是困难的：它们中没有一个允许交叉交换或平凡花瓣删除，即使允许两种自然的保持花瓣数的对称性也是如此。在这些对称性和镜像之下，有三个困难图。其中两个可以通过经过 11 个花瓣来解开；第三个无法通过最多 11 个花瓣的图解开，但可以通过 13 个花瓣解开。我们解释了这一现象与矩形情形不同的原因：花瓣图是一种弧表示，其页面的循环顺序由其在装订线上的顶点顺序决定，而 Cromwell 和 Dynnikov 的初等移动都不保持这种刚性结构。
英文摘要
A petal diagram of a knot is a projection with a single multi-crossing and no nested loops; it is encoded by a permutation of the heights of the strands through the multi-crossing. Colton, Glover, Hughes and Sandberg proved a Reidemeister-type theorem for petal diagrams: two petal permutations represent the same knot if and only if they are related by trivial petal additions and deletions and by crossing exchanges. We ask whether every petal diagram of the unknot can be reduced to the one-petal diagram without ever increasing the number of petals, in analogy with Dynnikov's monotonic simplification theorem for rectangular diagrams. By an exhaustive, certified computer search we show that this is true for diagrams with at most 7 petals and false for 9 petals. Of the 40320 petal diagrams with 9 petals, 24992 represent the unknot, and exactly 108 of them are hard: none of them admits a crossing exchange or a trivial petal deletion, even if two natural petal-number-preserving symmetries are allowed. Up to these symmetries and mirror image there are three hard diagrams. Two of them can be untangled by passing through 11 petals; the third cannot be untangled through diagrams with at most 11 petals, but can through 13. We explain why the phenomenon differs from the rectangular case: a petal diagram is an arc presentation whose cyclic order of pages is determined by the order of its vertices on the binding, and no elementary move of Cromwell and Dynnikov preserves this rigid structure.
