Thin position for 3-manifolds

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. …”

Circular thin position for knots inS3

“…Analogous to Scharlemann and Thompson [15], the following theorem holds: Theorem 3.2 If E.K/ is in circular thin position then:…”

“…R. This function corresponds to a handle decomposition of 3 , where b 0 is a collection of 0-handles, N i is a collection of 1-handles, T i is a collection of 2-handles and b 3 is a collection of 3-handles. In [15] Scharlemann and Thompson introduce the concept of thin position for 3-manifolds; the idea is to build the manifold as described before, with a sequence of 1-handles and 2-handles chosen to keep the boundaries of the intermediate steps as simple as possible.…”

Heegaard Genus of the Connected Sum of M-small Knots

“…Remark 2.5. It is well known that every irreducible Heegaard splitting of a 3-manifold M with incompressible boundary (other than surface cross interval) admits a S-T untelescoping such that each Heegaard splitting is non-trivial (see [25] and [24]). In this case, every component of ∂R j is incompressible in M. If in addition C 1 ∪ Σ C 2 is minimal genus then by [28] each component of ∂R j \ ∂M is an essential surface.…”