On critical Heegaard splittings of tunnel number two composite knot exteriors

TL;DR

This paper investigates conditions under which certain tunnel number two knots and their connected sums induce critical Heegaard splittings in their exteriors, providing new criteria and examples for such splittings.

Contribution

It establishes that specific weak reducing pairs induce critical Heegaard splittings in tunnel number two knot exteriors and characterizes when weak reducing pairs are uniquely determined by compressing disks.

Findings

Tunnel number two knots can induce critical Heegaard splittings under certain conditions.

Connected sums of 2-bridge and (1,1)-knots can produce critical Heegaard splittings.

A criterion for unique determination of weak reducing pairs in genus three, unstabilized splittings.

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.