The rectangle condition does not detect the strong irreducibility

Abstract

The rectangle condition for a genus $g$ Heegaard splitting of a 3-manifold, defined by Casson and Gordon, provides a sufficient criterion for the Heegaard splitting to be strongly irreducible. However it is unknown whether there exists a strongly irreducible Heegaard splitting which does not satisfy the rectangle condition. In this paper we provide a counterexample of a genus 2 Heegaard splitting of a 3-manifold which is strongly irreducible but fails to satisfy the rectangle condition. The way of constructing such an example is to take a double branched cover of a 3-bridge decomposition of a knot in $S^3$ which is strongly irreducible but does not meet the rectangle condition. This implies that the rectangle condition does not detect the strong irreducibility. As our next goal, we expect that this result provides the weaker version of the rectangle condition which detects the strong irreducibility.