Minimal partitions for critical Heegaard splittings

In this paper, we define the minimality of a partition for a critical Heegaard surface. The standard minimal genus Heegaard surface of [Formula: see text], which is known to be critical, admits a minimal partition. Moreover, we give an example of a critical surface that admits both a minimal partition and a non-minimal partition.

Rectangle condition for bridge decompositions of links

In this paper, we define the rectangle condition for bridge decompositions of links in [Formula: see text] whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of [Formula: see text]-manifolds. We show that the satisfaction of the rectangle condition for an [Formula: see text]-bridge decomposition can guarantee that the Hempel distance of the [Formula: see text]-bridge decomposition is greater than or equal to two.

A distance for circular Heegaard splittings

For a knot [Formula: see text], its exterior [Formula: see text] has a singular foliation by Seifert surfaces of [Formula: see text] derived from a circle-valued Morse function [Formula: see text]. When [Formula: see text] is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose [Formula: see text] into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of [Formula: see text]. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) [Formula: see text] cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for [Formula: see text] is unique up to isotopy.

2-Component links with genus two Heegaard splittings

In the present paper, we consider two types of 2-component links with genus two Heegaard splittings. One of them is an ordinary tunnel number one link, and the other is a somewhat different tunnel number one link. We will try to detect the differences between those two types. In fact, we will characterize composite tunnel number one links of the second type, and tunnel number one links of the second type with essential tori.