On critical Heegaard splittings of tunnel number two composite knot exteriors

Abstract

In this article, we prove that a tunnel number two knot induces a critical Heegaard splitting in its exterior if there are two weak reducing pairs such that each weak reducing pair contains the cocore disk of each tunnel. Moreover, we prove that a connected sum of two 2-bridge knots or more generally that of two $(1,1)$-knots can induce a critical Heegaard splitting in its exterior as the examples of the main theorem. Finally, we give an equivalent condition for a weak reducing pair to be determined by a compressing disk uniquely when the manifold is closed, irreducible and the Heegaard splitting is of genus three and unstabilized.