362 research outputs found
Knots and k-width
We investigate several integer invariants of curves in 3-space. We
demonstrate relationships of these invariants to crossing number and to total
curvature
Periodic orbits in the restricted three-body problem and Arnold's -invariant
We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman
systems. This is a class of Hamiltonian dynamical systems that describes the
dynamics of an electron in an external electric and magnetic field, and
includes many systems from celestial mechanics. Based on Arnold's
-invariant, we introduce invariants of periodic orbits in planar
Stark-Zeeman systems and study their behaviour.Comment: 36 Pages, 16 Figure
On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves
A long standing conjecture of Richter and Thomassen states that the total
number of intersection points between any simple closed Jordan curves in
the plane, so that any pair of them intersect and no three curves pass through
the same point, is at least .
We confirm the above conjecture in several important cases, including the
case (1) when all curves are convex, and (2) when the family of curves can be
partitioned into two equal classes such that each curve from the first class is
touching every curve from the second class. (Two curves are said to be touching
if they have precisely one point in common, at which they do not properly
cross.)
An important ingredient of our proofs is the following statement: Let be
a family of the graphs of continuous real functions defined on
, no three of which pass through the same point. If there are
pairs of touching curves in , then the number of crossing points is
.Comment: To appear in SODA 201
Maps on 3-manifolds given by surgery
Suppose that the 3-manifold M is given by integral surgery along a link L in
S^3. In the following we construct a stable map from M to the plane, whose
singular set is canonically oriented. We obtain upper bounds for the minimal
numbers of crossings and non-simple singularities and of connected components
of fibers of stable maps from M to the plane in terms of properties of L.Comment: 22 pages, 13 figures. Accepted for publication by Pacific J. Mat
Simple Riemannian surfaces are scattering rigid
Scattering rigidity of a Riemannian manifold allows one to tell the metric of
a manifold with boundary by looking at the directions of geodesics at the
boundary. Lens rigidity allows one to tell the metric of a manifold with
boundary from the same information plus the length of geodesics. There are a
variety of results about lens rigidity but very little is known for scattering
rigidity. We will discuss the subtle difference between these two types of
rigidities and prove that they are equivalent for two-dimensional simple
manifolds with boundaries. In particular, this implies that two-dimensional
simple manifolds (such as the flat disk) are scattering rigid since they are
lens/boundary rigid (Pestov--Uhlmann, 2005).Comment: 23 page
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