8,791 research outputs found
Combinatorial Sutured TQFT as Exterior Algebra
The idea of a sutured topological quantum field theory was introduced by
Honda, Kazez and Mati\'c (2008). A sutured TQFT associates a group to each
sutured surface and an element of this group to each dividing set on this
surface. The notion was originally introduced to talk about contact invariants
in Sutured Floer Homology. We provide an elementary example of a sutured TQFT,
which comes from taking exterior algebras of certain singular homology groups.
We show that this sutured TQFT coincides with that of Honda et al. using
-coefficients. The groups in our theory, being exterior algebras,
naturally come with the structure of a ring with unit. We give an application
of this ring structure to understanding tight contact structures on solid tori
Covering the Boundary of a Simple Polygon with Geodesic Unit Disks
We consider the problem of covering the boundary of a simple polygon on n
vertices using the minimum number of geodesic unit disks. We present an O(n
\log^2 n+k) time 2-approximation algorithm for finding the centers of the
disks, with k denoting the number centers found by the algorithm
Visually building Smale flows in S3
A Smale flow is a structurally stable flow with one dimensional invariant
sets. We use information from homology and template theory to construct,
visualize and in some cases, classify, nonsingular Smale flows in the 3-sphere
A 1-parameter family of spherical CR uniformizations of the figure eight knot complement
We describe a simple fundamental domain for the holonomy group of the
boundary unipotent spherical CR uniformization of the figure eight knot
complement, and deduce that small deformations of that holonomy group (such
that the boundary holonomy remains parabolic) also give a uniformization of the
figure eight knot complement. Finally, we construct an explicit 1-parameter
family of deformations of the boundary unipotent holonomy group such that the
boundary holonomy is twist-parabolic. For small values of the twist of these
parabolic elements, this produces a 1-parameter family of pairwise
non-conjugate spherical CR uniformizations of the figure eight knot complement
The Complexity of Separating Points in the Plane
We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles
New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems
We present a new data structure to approximate accurately and efficiently a
polynomial of degree given as a list of coefficients. Its properties
allow us to improve the state-of-the-art bounds on the bit complexity for the
problems of root isolation and approximate multipoint evaluation. This data
structure also leads to a new geometric criterion to detect ill-conditioned
polynomials, implying notably that the standard condition number of the zeros
of a polynomial is at least exponential in the number of roots of modulus less
than or greater than .Given a polynomial of degree with
for , isolating all its complex roots or
evaluating it at points can be done with a quasi-linear number of
arithmetic operations. However, considering the bit complexity, the
state-of-the-art algorithms require at least bit operations even for
well-conditioned polynomials and when the accuracy required is low. Given a
positive integer , we can compute our new data structure and evaluate at
points in the unit disk with an absolute error less than in
bit operations, where means
that we omit logarithmic factors. We also show that if is the absolute
condition number of the zeros of , then we can isolate all the roots of
in bit operations. Moreover, our
algorithms are simple to implement. For approximating the complex roots of a
polynomial, we implemented a small prototype in \verb|Python/NumPy| that is an
order of magnitude faster than the state-of-the-art solver \verb/MPSolve/ for
high degree polynomials with random coefficients
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