arXiv subjects
Publications and source records attributed to A. Anokhina.
Recently, for a limited class for bipartite links, the complicated Khovanov-Rozansky matrix factorization technique was reduced to an analogue of elementary Kauffman-Khovanov cycle calculus for an arbitrary $N$. In this note, we demonstrate the consistency of such reduction with the computation of the bipartite Khovanov polynomials for $N=2$. Namely, we explain how the Kauffman-Khovanov $2^2$-hypercube is shrinked to the bipartite 3-hypercube.
The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY-PT polynomials for any $N$ in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can be easily implemented as a computer program. Bipartite links form a rather large family including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY-PT polynomials using our developed generalized Goeritz method.
Bipartite calculus is a direct generalization of Kauffman planar expansion from $N=2$ to arbitrary $N$, applicable to the restricted class of knots which are entirely made of antiparallel lock tangles. Whenever applicable, it allows a straightforward generalization of the Khovanov calculus without a need of the technically complicated matrix factorization used for arbitrary $N$ in the Khovanov-Rozansky (KR) approach. The main object here is the $3^{n}$-dimensional hypercube with $n$ being the number of bipartite vertices. Maps, differentials, complex and Poincaré polynomials are straightforward and indeed reproduce the Khovanov-Rozansky polynomials in the known cases. This provides a great simplification of the Khovanov-Rozansky calculus on the bipartite locus, what can make it an accessible tool for the study of superpolynomials.
The recently suggested bipartite analysis extends the Kauffman planar decomposition to arbitrary $N$, i.e. extends it from the Jones polynomial to the HOMFLY polynomial. This provides a generic and straightforward non-perturbative calculus in an arbitrary Chern--Simons theory. Technically, this approach is restricted to knots and links which possess bipartite realizations, i.e. can be entirely glued from antiparallel lock (two-vertex) tangles rather than single-vertex $R$-matrices. However, we demonstrate that the resulting positive decomposition (PD), i.e. the representation of the fundamental HOMFLY polynomials as positive integer polynomials of the three parameters $ϕ$, $\barϕ$ and $D$, exists for arbitrary knots, not only bipartite ones. This poses new questions about the true significance of bipartite expansion, which appears to make sense far beyond its original scope, and its generalizations to higher representations. We have provided two explanations for the existence of the PD for non-bipartite knots. An interesting option is to resolve a particular bipartite vertex in a not-fully-bipartite diagram and reduce the HOMFLY polynomial to a linear combination of those for smaller diagrams. If the resulting diagrams correspond to bipartite links, this option provides a PD even to an initially non-bipartite knot. Another possibility for a non-bipartite knot is to have a bipartite clone with the same HOMFLY polynomial providing this PD. We also suggest a promising criterium for the existence of a bipartite realization behind a given PD, which is based on the study of the precursor Jones polynomials.
We generalize the recently discovered planar decomposition (Kauffman bracket) for the HOMFLY polynomials of bipartite knot/link diagrams to (anti)symmetrically colored HOMFLY polynomials. Cabling destroys planarity, but it is restored after projection to (anti)symmetric representations. This allows to go beyond arborescent calculus, which so far produced the majority of results for colored polynomials. Technicalities include combinations of projectors, and these can be handled rigorously, without any guess-work -- what can be also useful for other considerations, where reliable quantization was so far unavailable. We explicitly provide simple examples of calculation of the HOMFLY polynomials in symmetric representations with the use of our planar technique. These examples reveal what we call the bipartite evolution and the bipartite decomposition of squares of $\mathcal{R}$-matrices eigenvalues in the antiparallel channel.
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but for a special class of bipartite diagrams made entirely from the anitparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters -- and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov-Rozansky generalisations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.
We suggest a somewhat non-standard view on a set of curious, paradoxical from the standpoint of simple classical physics and everyday experience phenomena. There are the quantisation (discrete set of values) of the observables (e.g., energy, momentum, angular momentum); forbidden simultaneous measurements of the observables in the most cases (e.g., of a coordinate and momentum, of angular momentum projections on difference axis); counter-intuitive relations on the simultaneously measurable quantities (e.g., the famous expression for the square momentum $l(l+1)$ with the maximal projection $l$). These and other paradoxes are traditionally related to "purely quantum" phenomenon, i.e., having no analogue in the "classical world" ones. However, there are deep analogies between classical and "quantum" worlds, as soon as the quantum technique is applied to the classical phenomenon. We follow these analogies with the examples of relatively simple and well known models of classical physics, such as a simplified model of light transition through the media, a system of electric charges close to each other and far from the observer; the specific of motion in the Coulomb/Newtonian field. This text can be considered as a mini-course addressed to higher school and undergraduate students who are interested in basics of quantum mechanics, but are not yet ready for systematic study of standard courses. The text may be also useful to those who supervise such students.
We briefly summarise our results on jumps in the analytic formulas for the Khovanov(-Rozansky) polynomials. We conclude from the empiric data that there are ``regular'' and ``weird'' catastrophes, which drastically differ by form of the associated jumps in the Khovanov(-Rozansky) polynomials. This is the first step towards the catastrophe theory for the cohomological knot invariants. In particular, it can be another way to see these quantities as observables in cohomological quantum field theory.
We provide new evidence that the tangle calculus and "evolution" are applicable to the Khovanov polynomials for families of long braids inside the knot diagram. We show that jumps in evolution, peculiar for superpolynomials, are much less abundant than it was originally expected. Namely, for torus and twist satellites of a fixed companion knot, the main (most complicated) contribution does not jump, all jumps are concentrated in the torus and twist part correspondingly, where these jumps are necessary to make the Khovanov polynomial positive. Among other things, this opens a way to define a jump-free part of the colored Khovanov polynomials, which differs from the naive colored polynomial just "infinitesimally". The separation between jumping and smooth parts involves a combination of Rasmussen index and a new knot invariant, which we call "Thickness".
This paper describes a new $ν_e$ identification method specifically designed to improve the low-energy ($< 30\,\mathrm{GeV}$) $ν_e$ identification efficiency attained by enlarging the emulsion film scanning volume with the next generation emulsion readout system. A relative increase of 25-70% in the $ν_e$ low-energy region is expected, leading to improvements in the OPERA sensitivity to neutrino oscillations in the framework of the 3 + 1 model. The method is applied to a subset of data where the detection efficiency increase is expected to be more relevant, and one additional $ν_e$ candidate is found. The analysis combined with the $ν_τ$ appearance results improves the upper limit on $\sin^2 2θ_{μe}$ to 0.016 at 90% C.L. in the MiniBooNE allowed region $Δm^2_{41} \sim 0.3\,\mathrm{eV}^2$.
We compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum group decomposition for the HOMFLY polynomial of a satellite knot can be extended to the Khovanov polynomial, whose quantum group properties are not manifest. Namely, the Khovanov polynomial of a twisted Whitehead double or two-strand cable (the two simplest satellite families) can be presented as a naively deformed linear combination of the pattern and companion invariants. For a given companion, the satellite polynomial "smoothly" depends on the pattern but for the "jump" at one critical point defined by the s-invariant of the companion knot. A similar phenomenon is known for the knot Floer homology and tau-invariant for the same kind of satellites.
Dark photons are hypothetical massive vector particles that could mix with ordinary photons. The simplest theoretical model is fully characterised by only two parameters: the mass of the dark photon m$_{γ^{\mathrm{D}}}$ and its mixing parameter with the photon, $\varepsilon$. The sensitivity of the SHiP detector is reviewed for dark photons in the mass range between 0.002 and 10 GeV. Different production mechanisms are simulated, with the dark photons decaying to pairs of visible fermions, including both leptons and quarks. Exclusion contours are presented and compared with those of past experiments. The SHiP detector is expected to have a unique sensitivity for m$_{γ^{\mathrm{D}}}$ ranging between 0.8 and 3.3$^{+0.2}_{-0.5}$ GeV, and $\varepsilon^2$ ranging between $10^{-11}$ and $10^{-17}$.
An event topology with two secondary vertices compatible with the decay of short-lived particles was found in the analysis of neutrino interactions in the Opera target. The observed topology is compatible with tau neutrino charged current (CC) interactions with charm production and neutrino neutral current (NC) interactions with $c\overline{c}$ pair production. However, other processes can mimic this topology. A dedicated analysis was implemented to identify the underlying process. A Monte Carlo simulation was developed and complementary procedures were introduced in the kinematic reconstruction. A multivariate analysis technique was used to achieve an optimal separation of signal from background. Most likely, this event is a $ν_τ$ CC interaction with charm production, the tau and charm particle decaying into 1 prong and 2 prongs, respectively. The significance of this observation is evaluated.
We propose to build and operate a detector that, for the first time, will measure the process $pp\toνX$ at the LHC and search for feebly interacting particles (FIPs) in an unexplored domain. The TI18 tunnel has been identified as a suitable site to perform these measurements due to very low machine-induced background. The detector will be off-axis with respect to the ATLAS interaction point (IP1) and, given the pseudo-rapidity range accessible, the corresponding neutrinos will mostly come from charm decays: the proposed experiment will thus make the first test of the heavy flavour production in a pseudo-rapidity range that is not accessible by the current LHC detectors. In order to efficiently reconstruct neutrino interactions and identify their flavour, the detector will combine in the target region nuclear emulsion technology with scintillating fibre tracking layers and it will adopt a muon identification system based on scintillating bars that will also play the role of a hadronic calorimeter. The time of flight measurement will be achieved thanks to a dedicated timing detector. The detector will be a small-scale prototype of the scattering and neutrino detector (SND) of the SHiP experiment: the operation of this detector will provide an important test of the neutrino reconstruction in a high occupancy environment.
The OPERA experiment has conclusively observed the appearance of tau neutrinos in the muon neutrino CNGS beam. Exploiting the OPERA detector capabilities, it was possible to isolate high purity samples of $ν_{e}$, $ν_μ$ and $ν_τ$ charged current weak neutrino interactions, as well as neutral current weak interactions. In this Letter, the full dataset is used for the first time to test the three-flavor neutrino oscillation model and to derive constraints on the existence of a light sterile neutrino within the framework of the $3+1$ neutrino model. For the first time, tau and electron neutrino appearance channels are jointly used to test the sterile neutrino hypothesis. A significant fraction of the sterile neutrino parameter space allowed by LSND and MiniBooNE experiments is excluded at 90% C.L. In particular, the best-fit values obtained by MiniBooNE combining neutrino and antineutrino data are excluded at 3.3 $σ$ significance.
OPERA is a long-baseline experiment designed to search for $ν_μ\toν_τ$ oscillations in appearance mode. It was based at the INFN Gran Sasso laboratory (LNGS) and took data from 2008 to 2012 with the CNGS neutrino beam from CERN. After the discovery of $ν_τ$ appearance in 2015, with $5.1σ$ significance, the criteria to select $ν_τ$ candidates have been extended and a multivariate approach has been used for events identification. In this way the statistical uncertainty in the measurement of the oscillation parameters and of $ν_τ$ properties has been improved. Results are reported.
The OPERA experiment discovered muon neutrino into tau neutrino oscillations in appearance mode, detecting tau leptons by means of nuclear emulsion films. The apparatus was also endowed with electronic detectors with tracking capability, such as scintillator strips and resistive plate chambers. Because of its location, in the underground Gran Sasso laboratory, under 3800 m.w.e., the OPERA detector has also been used as an observatory for TeV muons produced by cosmic rays in the atmosphere. In this paper the measurement of the single muon flux modulation and of its correlation with the seasonal variation of the atmospheric temperature are reported.
The OPERA experiment has discovered the tau neutrino appearance in the CNGS muon neutrino beam, in agreement with the 3 neutrino flavour oscillation hypothesis. The OPERA neutrino interaction target, made of Emulsion Cloud Chamber, was particularly efficient in the reconstruction of electromagnetic showers. Moreover, thanks to the very high granularity of the emulsion films, showers induced by electrons can be distinguished from those induced by $π^0$s, thus allowing the detection of charged current interactions of electron neutrinos. In this paper the results of the search for electron neutrino events using the full dataset are reported. An improved method for the electron neutrino energy estimation is exploited. Data are compatible with the 3 neutrino flavour mixing model expectations and are used to set limits on the oscillation parameters of the 3+1 neutrino mixing model, in which an additional mass eigenstate $m_{4}$ is introduced. At high $Δm^{2}_{41}$ $( \gtrsim 0.1~\textrm{eV}^{2})$, an upper limit on $\sin^2 2θ_{μe}$ is set to 0.021 at 90% C.L. and $Δm^2_{41} \gtrsim 4 \times 10^{-3}~\textrm{eV}^{2}$ is excluded for maximal mixing in appearance mode.