42 research outputs found
Connected sums of knots and weakly reducible Heegaard splittings
This paper studies the question of whether minimal genus Heegaard splittings
of exterior spaces of knots which are connected sums are weakly reducible or
not. Furthermore it is shown that the Heegaard splittings of the knots used by
Morimoto to show that tunnel number can be sub-additive are all strongly
irreducible. These are the first examples of strongly irreducible minimal genus
Heegaard splittings of composite knots. We also give a characterization of when
is a set of primitive annuli on a handlebody simultaneously primitive. This
characterization is different from that given in [Go].Comment: 26 pages 10 figure
Heegaard Splittings of Twisted Torus Knots
Little is known on the classification of Heegaard splittings for hyperbolic
3-manifolds. Although Kobayashi gave a complete classification of Heegaard
splittings for the exteriors of 2-bridge knots, our knowledge of other classes
is extremely limited. In particular, there are very few hyperbolic manifolds
that are known to have a unique minimal genus splitting. Here we demonstrate
that an infinite class of hyperbolic knot exteriors, namely exteriors of
certain "twisted torus knots" originally studied by Morimoto, Sakuma and
Yokota, have a unique minimal genus Heegaard splitting of genus two. We also
conjecture that these manifolds possess irreducible yet weakly reducible
splittings of genus three. There are no known examples of such Heegaard
splittings.Comment: 4 pages 8 figure
High distance Heegaard splittings via fat train tracks
We define "fat" train tracks and use them to give a combinatorial criterion
for the Hempel distance of Heegaard splittings for closed orientable
3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on
knots in the three sphere.Comment: 25 pages no figures. to appear in Proceedings of "Knots Groups and
3-manifolds" Marseilles France 200