1,438 research outputs found
Bridge and pants complexities of knots
We modify an approach of Johnson to define the distance of a bridge splitting
of a knot in a 3-manifold using the dual curve complex and pants complex of the
bridge surface. This distance can be used to determine a complexity, which
becomes constant after a sufficient number of stabilizations and perturbations,
yielding an invariant of the manifold-knot pair. We also give evidence toward
the relationship between the pants distance of a bridge splitting and the
hyperbolic volume of the exterior of a knot.Comment: 34 pages, 12 figure
On the tree-width of knot diagrams
We show that a small tree-decomposition of a knot diagram induces a small
sphere-decomposition of the corresponding knot. This, in turn, implies that the
knot admits a small essential planar meridional surface or a small bridge
sphere. We use this to give the first examples of knots where any diagram has
high tree-width. This answers a question of Burton and of Makowsky and
Mari\~no.Comment: 14 pages, 6 figures. V2: Minor updates to expositio
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
Products of Farey graphs are totally geodesic in the pants graph
We show that for a surface S, the subgraph of the pants graph determined by
fixing a collection of curves that cut S into pairs of pants, once-punctured
tori, and four-times-punctured spheres is totally geodesic. The main theorem
resolves a special case of a conjecture made by Aramayona, Parlier, and
Shackleton and has the implication that an embedded product of Farey graphs in
any pants graph is totally geodesic. In addition, we show that a pants graph
contains a convex n-flat if and only if it contains an n-quasi-flat.Comment: v2: 25 pages, 16 figures. Completely rewritten, several figures added
for clarit
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