Reducing heegaard splittings

Additivity of tunnel number for small knots

“…We here investigate tunnel systems for K 1 # K 2 which correspond to weakly reducible Heegaard splittings and show how they define tunnel systems for K 1 and K 2 . We exploit the ideas in introduced in [2] and extended in [13] and [12], linking weakly reducible Heegaard splittings and incompressible surfaces. …”

“…The proof of the Main Theorem in [2] shows that if (V 1 , V 2 ) is irreducible but weakly reducible, then k ≥ 1 in any untelescoping of (V 1 , V 2 ).…”

Minimal hypersurfaces of least area

“…So there is a connected orientable stable minimal hypersurface S in M (ε) with area less than | Σ ε | = |Σ ε |. S could be equal to S, but this would imply that | Σ ε | > | S| which is not the case by (2). Thus, S is in the interior of M ε .…”

“…A compression body is a 3-manifold B with boundary with a particular boundary component ∂ + B = Σ × {0} such that B is obtained from Σ × [0, 1] by attaching 2-handles and 3-handles, where no attachments are performed along ∂ + B = Σ × {0} (see [2] for related definitions).…”

Knots With Arbitrarily High Distance Bridge Decompositions

“…About fifty years later, Moise proved in [22] that every closed orientable 3-manifold can admit a Heegaard splitting. It is then shown that certain properties of Heegaard splittings reflect topological characteristics of 3-manifolds: For example, any Heegaard splitting of a reducible 3-manifold is reducible [7], any irreducible Heegaard splitting of a non-Haken 3-manifold is strongly irreducible [4].…”