785 research outputs found
Thin position for knots in a 3-manifold
We extend the notion of thin multiple Heegaard splittings of a link in a
3-manifold to take into consideration not only compressing disks but also
cut-disks for the Heegaard surfaces. We prove that if H is a c-strongly
compressible bridge surface for a link K contained in a closed orientable
irreducible 3-manifold M then one of the following is satisfied:
H is stabilized,
H is meridionally stabilized,
H is perturbed, a component of K is removable, M contains an essential
meridional surface.Comment: 14 pages, 2 figure
C-essential surfaces in (3-manifold, graph) pairs
Let be a graph in a compact, orientable 3--manifold and let
be a subgraph. can be placed in bridge position with respect to a Heegaard
surface . We show that if is what we call -c-weakly
reducible in the complement of then either a "degenerate" situation occurs
or can be untelescoped and consolidated into a collection of "thick
surfaces" and "thin surfaces". The thin surfaces are c-essential
(c-incompressible and essential) in the graph exterior and each thick surface
is a strongly irreducible bridge surface in the complement of the thin
surfaces. This strengthens and extends previous results of Hayashi-Shimokawa
and Tomova to graphs in 3-manifolds that may have non-empty boundary.Comment: 33 pages, 9 figures. Accepted for publication in Communications in
Analysis and Geometr
- …