535 research outputs found
Alexander representation of tangles
A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on
the two disks in the boundary of the cylinder. Using an algebraic tool
developed by Lescop, we extend the Burau representation of braids to a functor
from the category of oriented tangles to the category of Z[t,t^{-1}]-modules.
For (1,1)-tangles (i.e., tangles with one endpoint on each disk) this invariant
coincides with the Alexander polynomial of the link obtained by taking the
closure of the tangle. We use the notion of plat position of a tangle to give a
constructive proof of invariance in this case.Comment: 13 pages, 5 figure
On a problem of A. Weil
A topological invariant of the geodesic laminations on a modular surface is
constructed. The invariant has a continuous part (the tail of a continued
fraction) and a combinatorial part (the singularity data). It is shown, that
the invariant is complete, i.e. the geodesic lamination can be recovered from
the invariant. The continuous part of the invariant has geometric meaning of a
slope of lamination on the surface.Comment: to appear Beitr\"age zur Algebra und Geometri
Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1
We prove that the hyperelliptic Torelli group is generated by Dehn twists about
separating curves that are preserved by the hyperelliptic involution. This verifies a
conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel
of the Burau representation evaluated at t = −1 and also the fundamental group of
the branch locus of the period mapping, and so we obtain analogous generating sets
for those. One application is that each component in Torelli space of the locus of
hyperelliptic curves becomes simply connected when curves of compact type are added
Cellular structure of -Brauer algebras
In this paper we consider the -Brauer algebra over a commutative
noetherian domain. We first construct a new basis for -Brauer algebras, and
we then prove that it is a cell basis, and thus these algebras are cellular in
the sense of Graham and Lehrer. In particular, they are shown to be an iterated
inflation of Hecke algebras of type Moreover, when is a field of
arbitrary characteristic, we determine for which parameters the -Brauer
algebras are quasi-heredity. So the general theory of cellular algebras and
quasi-hereditary algebras applies to -Brauer algebras. As a consequence, we
can determine all irreducible representations of -Brauer algebras by linear
algebra methods
Unusual Thermodynamics on the Fuzzy 2-Sphere
Higher spin Dirac operators on both the continuum sphere() and its fuzzy
analog() come paired with anticommuting chirality operators. A
consequence of this is seen in the fermion-like spectrum of these operators
which is especially true even for the case of integer-spin Dirac operators.
Motivated by this feature of the spectrum of a spin 1 Dirac operator on
, we assume the spin 1 particles obey Fermi-Dirac statistics. This
choice is inspite of the lack of a well defined spin-statistics relation on a
compact surface such as . The specific heats are computed in the cases of
the spin and spin 1 Dirac operators. Remarkably the specific heat
for a system of spin particles is more than that of the spin 1
case, though the number of degrees of freedom is more in the case of spin 1
particles. The reason for this is inferred through a study of the spectrums of
the Dirac operators in both the cases. The zero modes of the spin 1 Dirac
operator is studied as a function of the cut-off angular momentum and is
found to follow a simple power law. This number is such that the number of
states with positive energy for the spin 1 and spin system become
comparable. Remarks are made about the spectrums of higher spin Dirac operators
as well through a study of their zero-modes and the variation of their spectrum
with degeneracy. The mean energy as a function of temperature is studied in
both the spin and spin 1 cases. They are found to deviate from
the standard ideal gas law in 2+1 dimensions.Comment: 19 pages, 7 figures. The paper has been significantly modified. Main
results are unchange
Factoring Products of Braids via Garside Normal Form
Braid groups are infinite non-abelian groups naturally arising from geometric braids. For two decades they have been proposed for cryptographic use. In braid group cryptography public braids often contain secret braids as factors and it is hoped that rewriting the product of braid words hides individual factors. We provide experimental evidence that this is in general not the case and argue that under certain conditions parts of the Garside normal form of factors can be found in the Garside normal form of their product. This observation can be exploited to decompose products of braids of the form ABC when only B is known. Our decomposition algorithm yields a universal forgery attack on WalnutDSA™, which is one of the 20 proposed signature schemes that are being considered by NIST for standardization of quantum-resistant public-key cryptography. Our attack on WalnutDSA™ can universally forge signatures within seconds for both the 128-bit and 256-bit security level, given one random message-signature pair. The attack worked on 99.8% and 100% of signatures for the 128-bit and 256-bit security levels in our experiments. Furthermore, we show that the decomposition algorithm can be used to solve instances of the conjugacy search problem and decomposition search problem in braid groups. These problems are at the heart of other cryptographic schemes based on braid groups.SCOPUS: cp.kinfo:eu-repo/semantics/published22nd IACR International Conference on Practice and Theory of Public-Key Cryptography, PKC 2019; Beijing; China; 14 April 2019 through 17 April 2019ISBN: 978-303017258-9Volume Editors: Sako K.Lin D.Publisher: Springer Verla
Visualizing the Template of a Chaotic Attractor
Chaotic attractors are solutions of deterministic processes, of which the
topology can be described by templates. We consider templates of chaotic
attractors bounded by a genus-1 torus described by a linking matrix. This
article introduces a novel and unique tool to validate a linking matrix, to
optimize the compactness of the corresponding template and to draw this
template. The article provides a detailed description of the different
validation steps and the extraction of an order of crossings from the linking
matrix leading to a template of minimal height. Finally, the drawing process of
the template corresponding to the matrix is saved in a Scalable Vector Graphics
(SVG) file.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
In three spacetime dimensions, general relativity drastically simplifies,
becoming a ``topological'' theory with no propagating local degrees of freedom.
Nevertheless, many of the difficult conceptual problems of quantizing gravity
are still present. In this review, I summarize the rather large body of work
that has gone towards quantizing (2+1)-dimensional vacuum gravity in the
setting of a spatially closed universe.Comment: 61 pages, draft of review for Living Reviews; comments, criticisms,
additions, missing references welcome; v2: minor changes, added reference
Search for new phenomena in final states with an energetic jet and large missing transverse momentum in pp collisions at √ s = 8 TeV with the ATLAS detector
Results of a search for new phenomena in final states with an energetic jet and large missing transverse momentum are reported. The search uses 20.3 fb−1 of √ s = 8 TeV data collected in 2012 with the ATLAS detector at the LHC. Events are required to have at least one jet with pT > 120 GeV and no leptons. Nine signal regions are considered with increasing missing transverse momentum requirements between Emiss T > 150 GeV and Emiss T > 700 GeV. Good agreement is observed between the number of events in data and Standard Model expectations. The results are translated into exclusion limits on models with either large extra spatial dimensions, pair production of weakly interacting dark matter candidates, or production of very light gravitinos in a gauge-mediated supersymmetric model. In addition, limits on the production of an invisibly decaying Higgs-like boson leading to similar topologies in the final state are presente
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