 
Summary: REFILLING MERIDIANS IN A GENUS 2 HANDLEBODY
COMPLEMENT
MARTIN SCHARLEMANN
Dedicated to the memory of Heiner Zieschang, who first noticed that genus two
handlebodies could be interesting
ABSTRACT. Suppose a genus two handlebody is removed from a 3
manifold M and then a single meridian of the handlebody is restored.
The result is a knot or link complement in M and it is natural to ask
whether geometric properties of the link complement say something about
the meridian that was restored. Here we consider what the relation must
be between two not necessarily disjoint meridians so that restoring each
of them gives a trivial knot or a split link.
1. BACKGROUND
For a knot or link in a 3manifold, here are some natural geometric ques
tions that arise, in roughly ascending order of geometric sophistication: is
the knot the unknot? is the link split? is the link or knot a connected sum?
are there companion tori? beyond connected sums, are there essential an
nuli in the link complement? beyond connected sums, are there essential
meridional planar surfaces? One wellestablished context for such ques
tions is that of Dehn surgery (cf [Go]) where one imagines filling in the
