For a Heegaard splitting of a [Formula: see text]-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of [Formula: see text]-bridge decompositions of knots admitting a diagram with eight types of rectangles, which are not strongly irreducible. This says that the rectangle condition is sharp. Moreover, we define a variation of the rectangle condition so-called the sewing rectangle condition that also can guarantee the strong irreducibility of [Formula: see text]-bridge decompositions of knots. The new condition needs six types of rectangles but more complicated than nine types of rectangles for the rectangle condition.

中文翻译：

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Casson-Gordon 矩形条件的性质

对于 [Formula: see text]-流形的 Heegaard 分裂，Casson-Gordon 的矩形条件，简称为矩形条件，是其 Heegaard 图上的一个条件，可保证分裂的强不可约性；它需要九种类型的矩形来对应两侧的两条裤子的每种组合。矩形条件也适用于结的桥分解。我们给出了[公式：见文本]的例子——节的桥分解，允许一个包含八种矩形的图，这些矩形不是强不可约的。这表示矩形条件是尖锐的。此外，我们定义了一种矩形条件的变体，称为缝合矩形条件，它也可以保证结的[公式：见正文]-桥分解的强不可约性。