Horizontal Dehn surgery and distance of Heegaard splittings

(Joint with M. Lustig) Given a $3$ - manifold $M$ with a Heegaard surface $\Sigma$ of genus
$g \geq 2$ and an essential simple closed curve $c \subset \Sigma$, we
can obtain a new Heegaard splitting by changing the gluing of the two
handlebodies / compression bodies by a Dehn twist to some power $m$
along $c$. If $c$ is “sufficiently complicated”, measured a priori
by a parameter $n$, then there is at most a single value $m_0$ so that
the obtained Heegaard splitting is of smaller distance than $n -
1$. Furthermore the curves $c$ with this property are “generic” in
the set of essential simple closed curves $c \subset \Sigma$.