A generalization of the rectangle condition
Let \(M\) be a closed oriented \(3\)-manifold. If there is a closed surface \(S\) which cuts \(M\) into two handlebodies \(V\) and \(W\) with \(S=\partial_{+} V=\partial_{+} W\), then we say \(M\) has a Heegaard splitting, denoted by \(M=V\cup_{S} W\). \textit{A. Casson} and \textit{C. Gordon} [``Manifolds with irreducible Heegaard splittings of arbitrary large genus'', Unpublished] introduced the rectangle condition on Heegaard surfaces to show strong irreducibility of Heegaard splittings. In the paper under review, the author introduces the connecting rectangle condition and a general version of the rectangle condition to check whether or not \(M=V\cup_{S} W\) is strongly irreducible. Also, the author introduces the connecting rectangle condition on the \(n\)-bridge sphere for an \(n\)-bridge decomposition of a knot and checks whether or not the Hempel distance of an \(n\)-bridge decomposition is greater than or equal to two. Finally, the author gives an interesting example of an \(n\)-bridge decomposition of links which satisfy the connecting rectangle condition.
scientific article; zbMATH DE number 6882111