Heegaard splittings of distance exactly $n$
  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 $n$. 
Ayako Ido, Yeonhee Jang, Tsuyoshi Kobayashi
math.GT