278 research outputs found
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Area Inequalities for Embedded Disks Spanning Unknotted Curves
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric
inequality that bounds the area of an embedded disk spanning the curve in terms
of two parameters: the length L of the curve and the thickness r (maximal
radius of an embedded tubular neighborhood) of the curve. For fixed length, the
expression giving the upper bound on the area grows exponentially in 1/r^2. In
the direction of lower bounds, we give a sequence of length one curves with r
approaching 0 for which the area of any spanning disk is bounded from below by
a function that grows exponentially with 1/r. In particular, given any constant
A, there is a smooth, unknotted length one curve for which the area of a
smallest embedded spanning disk is greater than A.Comment: 31 pages, 5 figure
Man and machine thinking about the smooth 4-dimensional Poincar\'e conjecture
While topologists have had possession of possible counterexamples to the
smooth 4-dimensional Poincar\'{e} conjecture (SPC4) for over 30 years, until
recently no invariant has existed which could potentially distinguish these
examples from the standard 4-sphere. Rasmussen's s-invariant, a slice
obstruction within the general framework of Khovanov homology, changes this
state of affairs. We studied a class of knots K for which nonzero s(K) would
yield a counterexample to SPC4. Computations are extremely costly and we had
only completed two tests for those K, with the computations showing that s was
0, when a landmark posting of Akbulut (arXiv:0907.0136) altered the terrain.
His posting, appearing only six days after our initial posting, proved that the
family of ``Cappell--Shaneson'' homotopy spheres that we had geared up to study
were in fact all standard. The method we describe remains viable but will have
to be applied to other examples. Akbulut's work makes SPC4 seem more plausible,
and in another section of this paper we explain that SPC4 is equivalent to an
appropriate generalization of Property R (``in S^3, only an unknot can yield
S^1 x S^2 under surgery''). We hope that this observation, and the rich
relations between Property R and ideas such as taut foliations, contact
geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to
look at SPC4.Comment: 37 pages; changes reflecting that the integer family of
Cappell-Shaneson spheres are now known to be standard (arXiv:0907.0136
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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