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Martin Scharlemann

Publications and source records attributed to Martin Scharlemann.

41 records · Page 3Linked to original sources
arXiv1999

Annuli in generalized Heegaard splittings and degeneration of tunnel number

We analyze how a family of essential annuli in a compact 3-manifold will induce, from a strongly irreducible generalized Heegaard splitting of the ambient manifold, generalized Heegaard splittings of the complementary components. There are specific applications to the subadditivity of tunnel number of knots, improving somewhat bounds of Kowng. For example, in the absence of 2-bridge summands, the tunnel number of the sum of n knots is no less than 2/5 the sum of the tunnel numbers.

math.GT↗
arXiv1999

How a strongly irreducible Heegaard splitting intersects a handlebody

It has been shown elsewhere that a strongly irreducible Heegaard splitting surface Q of a compact orientable 3-manifold M can, under reasonable side conditions, intersect a ball or a solid torus in M in only a few possible ways. Here we extend those results to describe how Q can intersect a handlebody in M.

math.GT↗
arXiv1998

The structure of a solvmanifold's Heegaard splittings

We classify isotopy classes of irreducible Heegaard splittings of solvmanifolds. If the monodromy of the solvmanifold can be expressed as a 2 x 2 matrix with 0 in the lower right hand corner (as always is true when the absolute value of the trace is 3), then any irreducible splitting is strongly irreducible and of genus two. If furthermore the absolute value of the trace is 4 or greater, then any two such splittings are isotopic. If the absolute value of the trace is 3 then, up to isotopy, there are exactly two irreducible splittings, their associated hyperelliptic involutions commute, and the product of the involutions is the central involution of the solvmanifold. If the monodromy cannot be expressed as a 2 x 2 matrix with 0 in the lower right hand corner, then the splitting is weakly reducible, of genus three and unique up to isotopy.

math.GT↗
arXiv1998

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.

math.GT↗