3,461 research outputs found
On an action of the braid group B_{2g+2} on the free group F_{2g}
We construct an action of the braid group B_{2g+2} on the free group F_{2g}
extending an action of B_4 on F_2 introduced earlier by Reutenauer and the
author. Our action induces a homomorphism from B_{2g+2} into the symplectic
modular group Sp_{2g}(Z). In the special case g=2 we show that the latter
homomorphism is surjective and determine its kernel, thus obtaining a
braid-like presentation of Sp_4(Z).Comment: 11 pages. Minor changes in v
Mapping class group and U(1) Chern-Simons theory on closed orientable surfaces
U(1) Chern-Simons theory is quantized canonically on manifolds of the form
, where is a closed orientable surface. In
particular, we investigate the role of mapping class group of in the
process of quantization. We show that, by requiring the quantum states to form
representation of the holonomy group and the large gauge transformation group,
both of which are deformed by quantum effect, the mapping class group can be
consistently represented, provided the Chern-Simons parameter satisfies an
interesting quantization condition. The representations of all the discrete
groups are unique, up to an arbitrary sub-representation of the mapping class
group. Also, we find a duality of the representations.Comment: 17 pages, 3 figure
Braids of entangled particle trajectories
In many applications, the two-dimensional trajectories of fluid particles are
available, but little is known about the underlying flow. Oceanic floats are a
clear example. To extract quantitative information from such data, one can
measure single-particle dispersion coefficients, but this only uses one
trajectory at a time, so much of the information on relative motion is lost. In
some circumstances the trajectories happen to remain close long enough to
measure finite-time Lyapunov exponents, but this is rare. We propose to use
tools from braid theory and the topology of surface mappings to approximate the
topological entropy of the underlying flow. The procedure uses all the
trajectory data and is inherently global. The topological entropy is a measure
of the entanglement of the trajectories, and converges to zero if they are not
entangled in a complex manner (for instance, if the trajectories are all in a
large vortex). We illustrate the techniques on some simple dynamical systems
and on float data from the Labrador sea.Comment: 24 pages, 21 figures. PDFLaTeX with RevTeX4 macros. Matlab code
included with source. Fixed an inconsistent convention problem. Final versio
Stability of the magnetic Schr\"odinger operator in a waveguide
The spectrum of the Schr\"odinger operator in a quantum waveguide is known to
be unstable in two and three dimensions. Any enlargement of the waveguide
produces eigenvalues beneath the continuous spectrum. Also if the waveguide is
bent eigenvalues will arise below the continuous spectrum. In this paper a
magnetic field is added into the system. The spectrum of the magnetic
Schr\"odinger operator is proved to be stable under small local deformations
and also under small bending of the waveguide. The proof includes a magnetic
Hardy-type inequality in the waveguide, which is interesting in its own
Distance and intersection number in the curve graph of a surface
In this work, we study the cellular decomposition of induced by a filling
pair of curves and , , and its connection
to the distance function in the curve graph of a closed orientable
surface of genus . Efficient geodesics were introduced by the first
author in joint work with Margalit and Menasco in 2016, giving an algorithm
that begins with a pair of non-separating filling curves that determine
vertices in the curve graph of a closed orientable surface and
computing from them a finite set of {\it efficient} geodesics. We extend the
tools of efficient geodesics to study the relationship between distance
, intersection number , and . The main result is
the development and analysis of particular configurations of rectangles in
called \textit{spirals}. We are able to show that, in some
special cases, the efficient geodesic algorithm can be used to build an
algorithm that reduces while preserving . At the end of the
paper, we note a connection of our work to the notion of extending geodesics.Comment: 20 pages, 17 figures. Changes: A key lemma (Lemma 5.6) was revised to
be more precise, an irrelevant proposition (Proposition 2.1) and example were
removed, unnecessary background material was taken out, some of the
definitions and cited results were clarified (including added figures,) and
Proposition 5.7 and Theorem 5.8 have been merged into a single theorem,
Theorem 4.
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