1,005 research outputs found
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
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
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
Spectral Theory for Networks with Attractive and Repulsive Interactions
There is a wealth of applied problems that can be posed as a dynamical system
defined on a network with both attractive and repulsive interactions. Some
examples include: understanding synchronization properties of nonlinear
oscillator;, the behavior of groups, or cliques, in social networks; the study
of optimal convergence for consensus algorithm; and many other examples.
Frequently the problems involve computing the index of a matrix, i.e. the
number of positive and negative eigenvalues, and the dimension of the kernel.
In this paper we consider one of the most common examples, where the matrix
takes the form of a signed graph Laplacian. We show that the there are
topological constraints on the index of the Laplacian matrix related to the
dimension of a certain homology group. In certain situations, when the homology
group is trivial, the index of the operator is rigid and is determined only by
the topology of the network and is independent of the strengths of the
interactions. In general these constraints give upper and lower bounds on the
number of positive and negative eigenvalues, with the dimension of the homology
group counting the number of eigenvalue crossings. The homology group also
gives a natural decomposition of the dynamics into "fixed" degrees of freedom,
whose index does not depend on the edge-weights, and an orthogonal set of
"free" degrees of freedom, whose index changes as the edge weights change. We
also present some numerical studies of this problem for large random matrices.Comment: 27 pages; 9 Figure
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