1,255 research outputs found
Combinatorial complexity of signed discs
AbstractLet C+ and C− be two collections of topological discs. The collection of discs is ‘topological’ in the sense that their boundaries are Jordan curves and each pair of Jordan curves intersect at most twice. We prove that the region ∪C+ − ∪C− has combinatorial complexity at most 10n − 30 where p = |C+|, q = |C−| and n = p + q ≥ 5. Moreover, this bound is achievable. We also show less precise bounds that are stated as functions of p and q
Taut ideal triangulations of 3-manifolds
A taut ideal triangulation of a 3-manifold is a topological ideal
triangulation with extra combinatorial structure: a choice of transverse
orientation on each ideal 2-simplex, satisfying two simple conditions. The aim
of this paper is to demonstrate that taut ideal triangulations are very common,
and that their behaviour is very similar to that of a taut foliation. For
example, by studying normal surfaces in taut ideal triangulations, we give a
new proof of Gabai's result that the singular genus of a knot in the 3-sphere
is equal to its genus.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper12.abs.htm
A new algorithm for recognizing the unknot
The topological underpinnings are presented for a new algorithm which answers
the question: `Is a given knot the unknot?' The algorithm uses the braid
foliation technology of Bennequin and of Birman and Menasco. The approach is to
consider the knot as a closed braid, and to use the fact that a knot is
unknotted if and only if it is the boundary of a disc with a combinatorial
foliation. The main problems which are solved in this paper are: how to
systematically enumerate combinatorial braid foliations of a disc; how to
verify whether a combinatorial foliation can be realized by an embedded disc;
how to find a word in the the braid group whose conjugacy class represents the
boundary of the embedded disc; how to check whether the given knot is isotopic
to one of the enumerated examples; and finally, how to know when we can stop
checking and be sure that our example is not the unknot.Comment: 46 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol2/paper9.abs.htm
An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks
We present an exact and complete algorithm to isolate the real solutions of a
zero-dimensional bivariate polynomial system. The proposed algorithm
constitutes an elimination method which improves upon existing approaches in a
number of points. First, the amount of purely symbolic operations is
significantly reduced, that is, only resultant computation and square-free
factorization is still needed. Second, our algorithm neither assumes generic
position of the input system nor demands for any change of the coordinate
system. The latter is due to a novel inclusion predicate to certify that a
certain region is isolating for a solution. Our implementation exploits
graphics hardware to expedite the resultant computation. Furthermore, we
integrate a number of filtering techniques to improve the overall performance.
Efficiency of the proposed method is proven by a comparison of our
implementation with two state-of-the-art implementations, that is, LPG and
Maple's isolate. For a series of challenging benchmark instances, experiments
show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201
Intersection of paraboloids and application to Minkowski-type problems
In this article, we study the intersection (or union) of the convex hull of N
confocal paraboloids (or ellipsoids) of revolution. This study is motivated by
a Minkowski-type problem arising in geometric optics. We show that in each of
the four cases, the combinatorics is given by the intersection of a power
diagram with the unit sphere. We prove the complexity is O(N) for the
intersection of paraboloids and Omega(N^2) for the intersection and the union
of ellipsoids. We provide an algorithm to compute these intersections using the
exact geometric computation paradigm. This algorithm is optimal in the case of
the intersection of ellipsoids and is used to solve numerically the far-field
reflector problem
Unsigned state models for the Jones polynomial
It is well a known and fundamental result that the Jones polynomial can be
expressed as Potts and vertex partition functions of signed plane graphs. Here
we consider constructions of the Jones polynomial as state models of unsigned
graphs and show that the Jones polynomial of any link can be expressed as a
vertex model of an unsigned embedded graph.
In the process of deriving this result, we show that for every diagram of a
link in the 3-sphere there exists a diagram of an alternating link in a
thickened surface (and an alternating virtual link) with the same Kauffman
bracket. We also recover two recent results in the literature relating the
Jones and Bollobas-Riordan polynomials and show they arise from two different
interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric
On the Fukaya Categories of Higher Genus Surfaces
We construct the Fukaya category of a closed surface equipped with an area
form using only elementary (essentially combinatorial) methods. We also compute
the Grothendieck group of its derived category.Comment: 40 pages, 17 figures. Final Versio
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