8,759 research outputs found

    Products of Farey graphs are totally geodesic in the pants graph

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    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

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    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 H2×R\mathbb{H}^2 \times \mathbb{R}

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    We construct the first examples of complete, properly embedded minimal surfaces in H2×R\mathbb{H}^2 \times \mathbb{R} 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

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    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

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    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 Rd\R^d with a uniformly-chosen root is dd-parabolic. We then derive few geometric corollaries. E.g.\,every infinite graph packed in Rd\R^{d} has either strictly positive isoperimetric Cheeger constant or admits arbitrarily large finite sets WW with boundary size which satisfies ∣∂W∣≤∣W∣d−1d+o(1) |\partial W| \leq |W|^{\frac{d-1}{d}+o(1)}. Some open problems and conjectures are gathered at the end
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