Abstract: In this talk, we define the rectangle condition and connecting rectangle condition to check whether or not a Heegaard splitting is strongly irreducible which is a variation of the rectangle condition by Casson and Gordon. Moreover, with a similar condition defined on a n-bridge decomposition, we can check whether or not the Hempel distance of a n-bridge decomposition is greater than or equal to two. Then, we can define a generalized rectangle condition. We also give specific examples for Heegaard splitting of 3-manifolds and n-bridge decomposition of knots which satisfy the connecting rectangle condition.
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