1,553 research outputs found
Trisections and spun 4-manifolds
We study trisections of 4-manifolds obtained by spinning and twist-spinning
3-manifolds, and we show that, given a (suitable) Heegaard diagram for the
3-manifold, one can perform simple local modifications to obtain a trisection
diagram for the 4-manifold. We also show that this local modification can be
used to convert a (suitable) doubly-pointed Heegaard diagram for a
3-manifold/knot pair into a doubly-pointed trisection diagram for the
4-manifold/2-knot pair resulting from the twist-spinning operation.
This technique offers a rich list of new manifolds that admit trisection
diagrams that are amenable to study. We formulate a conjecture about
4-manifolds with trisection genus three and provide some supporting evidence.Comment: 16 pages, 12 figures. Comments welcome
Fibered ribbon disks
We study the relationship between fibered ribbon 1-knots and fibered ribbon
2-knots by studying fibered slice disks with handlebody fibers. We give a
characterization of fibered homotopy-ribbon disks and give analogues of the
Stallings twist for fibered disks and 2-knots. As an application, we produce
infinite families of distinct homotopy-ribbon disks with homotopy equivalent
exteriors, with potential relevance to the Slice-Ribbon Conjecture. We show
that any fibered ribbon 2-knot can be obtained by doubling infinitely many
different slice disks (sometimes in different contractible 4-manifolds).
Finally, we illustrate these ideas for the examples arising from spinning
fibered 1-knots.Comment: 20 pages, 3 figures. Version two has improved exposition and
incorporates referee suggestions. This version has been accepted for
publicatio
Bridge trisections of knotted surfaces in 4--manifolds
We prove that every smoothly embedded surface in a 4--manifold can be
isotoped to be in bridge position with respect to a given trisection of the
ambient 4--manifold; that is, after isotopy, the surface meets components of
the trisection in trivial disks or arcs. Such a decomposition, which we call a
\emph{generalized bridge trisection}, extends the authors' definition of bridge
trisections for surfaces in . Using this new construction, we give
diagrammatic representations called \emph{shadow diagrams} for knotted surfaces
in 4--manifolds. We also provide a low-complexity classification for these
structures and describe several examples, including the important case of
complex curves inside . Using these examples, we prove that
there exist exotic 4--manifolds with --trisections for certain values of
. We conclude by sketching a conjectural uniqueness result that would
provide a complete diagrammatic calculus for studying knotted surfaces through
their shadow diagrams.Comment: 17 pages, 5 figures. Comments welcom
Characterizing Dehn surgeries on links via trisections
We summarize and expand known connections between the study of Dehn surgery
on links and the study of trisections of closed, smooth 4-manifolds. In
addition, we describe how the potential counterexamples to the Generalized
Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus
four trisections of the standard four-sphere that are unlikely to be standard.
Finally, we give an analog of the Casson- Gordon Rectangle Condition for
trisections that can be used to obstruct reducibility of a given trisection.Comment: 15 pages, 4 color figures. Comments welcome
- …