138 research outputs found
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
On the tree-width of knot diagrams
We show that a small tree-decomposition of a knot diagram induces a small
sphere-decomposition of the corresponding knot. This, in turn, implies that the
knot admits a small essential planar meridional surface or a small bridge
sphere. We use this to give the first examples of knots where any diagram has
high tree-width. This answers a question of Burton and of Makowsky and
Mari\~no.Comment: 14 pages, 6 figures. V2: Minor updates to expositio
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