1,086 research outputs found
Heegaard genus formula for Haken manifolds
Given a 3-manifold M containing an incompressible surface Q, we obtain an
inequality relating the Heegaard genus of M and the Heegaard genera of the
components of M - Q. Here the sum of the genera of the components of M - Q is
bounded above by a linear expression in terms of the genus of M, the Euler
characteristic of Q and the number of parallelism classes of essential annuli
for which representatives can be simultaneously imbedded in the components of M
- Q.Comment: 21 pages, 17 figure
Comparing Heegaard and JSJ structures of orientable 3-manifolds
The Heegaard genus g of an irreducible closed orientable 3-manifold puts a
limit on the number and complexity of the pieces that arise in the
Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For
example, if p of the complementary components are not Seifert fibered, then p <
g. This result generalizes work of Kobayashi. The Heegaard genus g also puts
explicit bounds on the complexity of the Seifert pieces. For example, if the
union of the base spaces of the Seifert pieces has Euler characteristic X and
there are a total of f exceptional fibers in the Seifert pieces, then f - X is
no greater than 3g - 3 - p.Comment: 30 pages, 10 figure
The tunnel number of the sum of n knots is at least n
We prove that the tunnel number of the sum of n knots is at least n.Comment: 8 pages. To appear in Topolog
Contractibility of the Kakimizu complex and symmetric Seifert surfaces
Kakimizu complex of a knot is a flag simplicial complex whose vertices
correspond to minimal genus Seifert surfaces and edges to disjoint pairs of
such surfaces. We discuss a general setting in which one can define a similar
complex. We prove that this complex is contractible, which was conjectured by
Kakimizu. More generally, the fixed-point set (in the Kakimizu complex) for any
subgroup of an appropriate mapping class group is contractible or empty.
Moreover, we prove that this fixed-point set is non-empty for finite subgroups,
which implies the existence of symmetric Seifert surfaces.Comment: 24 pages, 7 figure
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