We expect manifolds obtained by Dehn filling to inherit properties from
the knot manifold. To what extent does that hold true for the Heegaard
structure? We study four changes to the Heegaard structure that may occur
after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface
is created, (3) a non-stabilized Heegaard surface destabilizes, and (4)
two or more non-isotopic Heegaard surfaces become isotopic.
We survey general results that give quite satisfactory restrictions
to phenomena (1) and (2) and, in a parallel thread, give a complete
classification of when all four phenomena occur when filling most
torus knot exteriors. This latter thread yields sufficient (and perhaps
necessary) conditions for the occurrence of phenomena (3) and (4).
