Heegaard distance of the link complements in S3

We show that, for any integers, [Formula: see text] and [Formula: see text], there exists a link in [Formula: see text] such that its complement has a genus [Formula: see text] Heegaard splitting with distance [Formula: see text].

Many Haken Heegaard splittings

We give a simple criterion for a Heegaard splitting to yield a Haken manifold. As a consequence, we construct many Haken manifolds, in particular homology spheres, with prescribed properties, namely Heegaard genus, Heegaard distance and Casson invariant. Along the way we give simpler and shorter proofs of the existence of splittings with specified Heegaard distance, originally proven by Ido–Jang–Kobayashi, of the existence of hyperbolic manifolds with prescribed Casson invariant, originally due to Lubotzky–Maher–Wu, and of a result about subsurface projections of disc sets (for which we even get better constants), originally due to Masur–Schleimer.

Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum

Let $M_{i}$ be a compact orientable 3-manifold, and $A_{i}$ an incompressible annulus on a component $F_i$ of $\partial M_i$, $i=1,2$. Suppose $A_{1}$ is separating on $F_{1}$ and $A_{2}$ is non-separating on $F_{2}$. Let $M$ be the annulus sum of $M_1$ and $M_2$ along $A_1$ and $A_2$. In the present paper we show that if $M_{i}$ has a Heegaard splitting $V_{i}\cup_{S_{i}}W_{i}$ with Heegaard distance $d(S_{i})\geq2g(M_{i})+5$ for $i=1,2$, then $g(M)=g(M_{1})+g(M_{2})$. Moreover, when $g(F_{2})\geq 2$, the minimal Heegaard splitting of $M$ is unique up to isotopy.

Small 3-manifolds with large Heegaard distance

We construct examples of closed non-Haken hyperbolic 3-manifolds with a Heegaard splitting of arbitrarily large distance.