Heegaard splittings of distance exactly nn

In this paper, we show that, for any integers $n\geq 2$ and $g\geq 2$, there exist genus-$g$ Heegaard splittings of compact 3-manifolds with distance exactly $n$.Comment: 13 pages, 2 figure

On keen bridge splittings of links

In this paper, we extend the concept of {\it (strongly) keenness} for Heegaard splittings to bridge splittings, and show that, for any integers $g$, $b$ and $n$ with $g\ge 0$, $b\ge 1$, $n\ge 1$ except for $(g,b)=(0,1)$ and $(g,b,n)=(0,3,1)$, there exists a strongly keen $(g,b)$-splitting of a link with distance $n$. We also show that any $(0,3)$-splitting of a link with distance $1$ cannot be keen