Properties of Casson-Gordon's rectangle condition

For a Heegaard splitting of a 3-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 3-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 3-bridge decompositions of knots. The new condition needs six types of rectangles but more complicated than nine types of rectangles for the rectangle condition.