2,782 research outputs found
Braids: A Survey
This article is about Artin's braid group and its role in knot theory. We set
ourselves two goals: (i) to provide enough of the essential background so that
our review would be accessible to graduate students, and (ii) to focus on those
parts of the subject in which major progress was made, or interesting new
proofs of known results were discovered, during the past 20 years. A central
theme that we try to develop is to show ways in which structure first
discovered in the braid groups generalizes to structure in Garside groups,
Artin groups and surface mapping class groups. However, the literature is
extensive, and for reasons of space our coverage necessarily omits many very
interesting developments. Open problems are noted and so-labelled, as we
encounter them.Comment: Final version, revised to take account of the comments of readers. A
review article, to appear in the Handbook of Knot Theory, edited by W.
Menasco and M. Thistlethwaite. 91 pages, 24 figure
Strong contraction of the representations of the three dimensional Lie algebras
For any Inonu-Wigner contraction of a three dimensional Lie algebra we
construct the corresponding contractions of representations. Our method is
quite canonical in the sense that in all cases we deal with realizations of the
representations on some spaces of functions; we contract the differential
operators on those spaces along with the representation spaces themselves by
taking certain pointwise limit of functions. We call such contractions strong
contractions. We show that this pointwise limit gives rise to a direct limit
space. Many of these contractions are new and in other examples we give a
different proof
Abelian covers of surfaces and the homology of the level L mapping class group
We calculate the first homology group of the mapping class group with
coefficients in the first rational homology group of the universal abelian -cover of the surface. If the surface has one marked point, then the
answer is \Q^{\tau(L)}, where is the number of positive divisors of
. If the surface instead has one boundary component, then the answer is
\Q. We also perform the same calculation for the level subgroup of the
mapping class group. Set . If the surface has one
marked point, then the answer is \Q[H_L], the rational group ring of .
If the surface instead has one boundary component, then the answer is \Q.Comment: 32 pages, 10 figures; numerous corrections and simplifications; to
appear in J. Topol. Ana
Hyperspherical theory of anisotropic exciton
A new approach to the theory of anisotropic exciton based on Fock
transformation, i.e., on a stereographic projection of the momentum to the unit
4-dimensional (4D) sphere, is developed. Hyperspherical functions are used as a
basis of the perturbation theory. The binding energies, wave functions and
oscillator strengths of elongated as well as flattened excitons are obtained
numerically. It is shown that with an increase of the anisotropy degree the
oscillator strengths are markedly redistributed between optically active and
formerly inactive states, making the latter optically active. An approximate
analytical solution of the anisotropic exciton problem taking into account the
angular momentum conserving terms is obtained. This solution gives the binding
energies of moderately anisotropic exciton with a good accuracy and provides a
useful qualitative description of the energy level evolution.Comment: 23 pages, 8 figure
The many faces of Lorenz knots
One of the greatest pleasures in doing mathematics (and one of the surest signs of being onto something really relevant) is discovering that two apparently completely unrelated objects actually are one and the same thing. This is what Étienne Ghys, of the École Normale Superieure de Lyon, did a few years ago (see [1] for the technical details), showing that the class of Lorenz knots, pertaining to the theory of chaotic dynamical systems and ordinary differential equations, and the class of modular knots, pertaining to the theory of 2-dimensional lattices and to number theory, coincide. In this short note we shall try to explain what Lorenz and modular knots are, and to give a hint of why they are the same. See also [2] for a more detailed but still accessible presentation, containing the beautiful pictures and animations prepared by Jos Leys [3], a digital artist, to illustrate Ghys’ results
Invariants of 2+1 Quantum Gravity
In [1,2] we established and discussed the algebra of observables for 2+1
gravity at both the classical and quantum level. Here our treatment broadens
and extends previous results to any genus with a systematic discussion of
the centre of the algebra. The reduction of the number of independent
observables to is treated in detail with a precise
classification for and .Comment: 10 pages, plain TEX, no figures, DFTT 46/9
On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2
Let denote the negative eigenvalues of the one-dimensional
Schr\"odinger operator on . We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb
R} V^{\gamma+1/2}(x)dx, (1) for the "limit" case This will imply
improved estimates for the best constants in (1), as
$1/2<\gamma<3/2.Comment: AMS-LATEX, 15 page
Every mapping class group is generated by 6 involutions
Let Mod_{g,b} denote the mapping class group of a surface of genus g with b
punctures. Feng Luo asked in a recent preprint if there is a universal upper
bound, independent of genus, for the number of torsion elements needed to
generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements
suffice to generate Mod_{g,0}. We also prove the more delicate result that
there is an upper bound, independent of genus, not only for the number of
torsion elements needed to generate Mod_{g,b} but also for the order of those
elements. In particular, our main result is that 6 involutions (i.e.
orientation-preserving diffeomorphisms of order two) suffice to generate
Mod_{g,b} for every genus g >= 3, b = 0, and g >= 4, b = 1.Comment: 15 pages, 7 figures; slightly improved main result; minor revisions.
to appear in J. Al
Moment operators of the Cartesian margins of the phase space observables
The theory of operator integrals is used to determine the moment operators of
the Cartesian margins of the phase space observables generated by the mixtures
of the number states. The moments of the -margin are polynomials of the
position operator and those of the -margin are polynomials of the momentum
operator.Comment: 14 page
- …