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.

18 pages, 11 figures

References in corpus (1)

• On unstabilzed genus three critical Heegaard surfaces

Cited by in corpus (5)

• A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical
• On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three I - the Structure Theorem
• The tunnel number and the cutting number with constituent handlebody-knots
• Generalized Heegaard splittings and the disk complex
• On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three III - Generalized Heegaard splittings and mapping classes