59,816 research outputs found
Simplicial embeddings between multicurve graphs
We study some graphs associated to a surface, called k-multicurve graphs,
which interpolate between the curve complex and the pants graph. Our main
result is that, under certain conditions, simplicial embeddings between
multicurve graphs are induced by -injective embeddings of the
corresponding surfaces. We also prove the rigidity of the multicurve graphs.Comment: New introduction and some changes in Section 2, main results
unchanged. References added. 18 pages, 5 figure
Avoiding the Global Sort: A Faster Contour Tree Algorithm
We revisit the classical problem of computing the \emph{contour tree} of a
scalar field , where is a
triangulated simplicial mesh in . The contour tree is a
fundamental topological structure that tracks the evolution of level sets of
and has numerous applications in data analysis and visualization.
All existing algorithms begin with a global sort of at least all critical
values of , which can require (roughly) time. Existing
lower bounds show that there are pathological instances where this sort is
required. We present the first algorithm whose time complexity depends on the
contour tree structure, and avoids the global sort for non-pathological inputs.
If denotes the set of critical points in , the running time is
roughly , where is the depth of in
the contour tree. This matches all existing upper bounds, but is a significant
improvement when the contour tree is short and fat. Specifically, our approach
ensures that any comparison made is between nodes in the same descending path
in the contour tree, allowing us to argue strong optimality properties of our
algorithm.
Our algorithm requires several novel ideas: partitioning in
well-behaved portions, a local growing procedure to iteratively build contour
trees, and the use of heavy path decompositions for the time complexity
analysis
Khovanov's homology for tangles and cobordisms
We give a fresh introduction to the Khovanov Homology theory for knots and
links, with special emphasis on its extension to tangles, cobordisms and
2-knots. By staying within a world of topological pictures a little longer than
in other articles on the subject, the required extension becomes essentially
tautological. And then a simple application of an appropriate functor (a
`TQFT') to our pictures takes them to the familiar realm of complexes of
(graded) vector spaces and ordinary homological invariants.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper33.abs.htm
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
Nielsen Realisation by Gluing: Limit Groups and Free Products
We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation
theorems from the setting of free groups to that of free products. As a result,
we obtain a fixed point theorem for finite groups of outer automorphisms acting
on the relative free splitting complex of Handel--Mosher and on the outer space
of a free product of Guirardel--Levitt, as well as a relative version of the
Nielsen realisation theorem, which in the case of free groups answers a
question of Karen Vogtmann. We also prove Nielsen realisation for limit groups,
and as a byproduct obtain a new proof that limit groups are CAT(). The
proofs rely on a new version of Stallings' theorem on groups with at least two
ends, in which some control over the behaviour of virtual free factors is
gained.Comment: 28 pages, 1 figur
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