Parity condition for irreducibility of Heegaard splittings

Heegaard 分解强不可约性的图解判据与 Goeritz 群的有限性

Diagrammatic criteria for strong irreducibility of Heegaard splittings and finiteness of Goeritz groups

本文提出两个基于任意圆盘系统（包括极小圆盘系统）的 Heegaard 图解判据，分别保证分解的强不可约性和 Goeritz 群的有限性，推广了 Casson-Gordon 与 Lustig-Moriah 的结果。

矩形条件无法检测强不可约性

The rectangle condition does not detect the strong irreducibility

本文通过构造 $S^3$ 中纽结 3-桥分解的二重分支覆盖作为反例，证明了 Casson 和 Gordon 的矩形条件无法检测 3-流形 Heegaard 分裂的强不可约性。

Parity criterion and Dehn twists for unstabilized Heegaard splittings

We give a parity condition of a Heegaard diagram to show that it is unstabilized. This improves the result of [5]. As an application, we construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.

Rectangle condition for compression body and 2-fold branched covering

We give the rectangle condition for strong irreducibility of Heegaard splittings of $3$-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of a $2$-fold covering of $S^3$ branched along a link. The condition implies that any thin meridional level surface in the link complement is incompressible. We also show that the additivity of knot width holds for a composite knot satisfying the condition.