8,759 research outputs found
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
The totally nonnegative Grassmannian is a ball
We prove that three spaces of importance in topological combinatorics are
homeomorphic to closed balls: the totally nonnegative Grassmannian, the
compactification of the space of electrical networks, and the cyclically
symmetric amplituhedron.Comment: 19 pages. v2: Exposition improved in many place
Minimal surfaces with positive genus and finite total curvature in
We construct the first examples of complete, properly embedded minimal
surfaces in with finite total curvature and
positive genus. These are constructed by gluing copies of horizontal catenoids
or other nondegenerate summands. We also establish that every horizontal
catenoid is nondegenerate.
Finally, using the same techniques, we are able to produce properly embedded
minimal surfaces with infinitely many ends. Each annular end has finite total
curvature and is asymptotic to a vertical totally geodesic plane.Comment: 32 pages, 4 figures. This revised version will appear in Geometry and
Topolog
Totally Twisted Khovanov Homology
We define a variation of Khovanov homology with an explicit description in
terms of the spanning trees of a link projection. We prove that this new theory
is a link invariant and describe some of its properties. Finally, we provide
some the results of some computer computations of the invariant.Comment: 45 pages, 21 figure
On limits of Graphs Sphere Packed in Euclidean Space and Applications
The core of this note is the observation that links between circle packings
of graphs and potential theory developed in \cite{BeSc01} and \cite{HS} can be
extended to higher dimensions. In particular, it is shown that every limit of
finite graphs sphere packed in with a uniformly-chosen root is
-parabolic. We then derive few geometric corollaries. E.g.\,every infinite
graph packed in has either strictly positive isoperimetric Cheeger
constant or admits arbitrarily large finite sets with boundary size which
satisfies . Some open problems and
conjectures are gathered at the end
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