arXiv research

Hashing is a basic tool for dimensionality reduction employed in several aspects of machine learning. However, the perfomance analysis is often carried out under the abstract assumption that a truly random unit cost hash function is used, without concern for which concrete hash function is employed. The concrete hash f…

A virtual string can be defined as a closed curve on a surface modulo certain equivalence relations. Turaev defined several invariants of virtual strings which we use to produce a table of virtual strings up to 4 crossings. We discuss progress in extending the tabulation to 5 crossings. We also provide a counter-exampl…

This paper tabulates prime knot projections up to eight double points.

We introduce a new way to tabulate knots by representing knot diagrams using a pair of planar trees. This pair of trees have their edges labeled by integers, they have no valence 2 vertices, and they have the same number of valence 1 vertices. The number of valence 1 vertices of the trees is called the girth of the kno…

Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces $L(p,q)$. For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein mod…

BPt is a Python library for ML with neuroimaging data.

The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.

Modern reinforcement learning algorithms reach super-human performance on many board and video games, but they are sample inefficient, i.e. they typically require significantly more playing experience than humans to reach an equal performance level. To improve sample efficiency, an agent may build a model of the enviro…

The paper improves bounds on knot crossings and tabulates minimal diagrams.

We model anomaly and change in data by embedding the data in an ultrametric space. Taking our initial data as cross-tabulation counts (or other input data formats), Correspondence Analysis allows us to endow the information space with a Euclidean metric. We then model anomaly or change by an induced ultrametric. The in…

The paper calculates minimal ribbonlength for various knots.

As a supplement to the authors' article "Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots", to appear in the Journal of Knot Theory and its Ramifications, we present minimal arc presentations of the prime knots up to arc index 11.

The paper addresses the $k$-tangle enumeration problem. We introduce a notion of cascade diagram for $k$-tangle projections. An effective enumeration algorithm for projections is proposed based on cascade representation. Tangles projections with up to 12 crossings are tabulated. We provide also pictures of alternating …

We present the results of Axel Seeliger's tabulation of symmetric union presentations for ribbon knots with crossing numbers 11 and 12 and exhibit possible examples for ribbon knots which are not representable as symmetric unions. In addition, we give a complete atlas of band diagrams for prime ribbon knots with 11 and…

We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately …

Listed 19,513 prime knots with arc index 12-16.

A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By the construction, $F$-polynomials are generalizations of the Kauffman's Affine …

Gray & Hervella gave a classification of almost Hermitian structures (g,I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g,I,J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we ob…

We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last re…

We consider a natural model of random knotting- choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to compute exact probabilities for knots in this model. As expected, most diagrams with 1…

Classifies prime algebraic tangles up to 14 crossings.

It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the clos…

Classifies uncolored bonded knots with up to 7 singularity points.

Enumerates knots up to five crossings and describes moves between them.

Study of group orderability using tensor and exterior squares.

Generative Adversarial Networks (GANs) and their extensions have carved open many exciting ways to tackle well known and challenging medical image analysis problems such as medical image de-noising, reconstruction, segmentation, data simulation, detection or classification. Furthermore, their ability to synthesize imag…

This is a PhD thesis about low dimensional topology, in particular knot thory in 3-manifolds also different from the 3-sphere, topological applications of quantum invariants, and Turaev's shadows. There is an introduction and a survey for these topics. The thesis uses skein theory and focues on the connected sum of cop…

Principled mapping from pure-DP ε to GDP μ for Gaussian differential privacy

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric …

Examples are given of prime Legendrian knots in the standard contact 3-space that have arbitrarily many distinct Chekanov polynomials, refuting a conjecture of Lenny Ng. These are constructed using a new `Legendrian tangle replacement' technique. This technique is then used to show that the phenomenon of multiple Cheka…

In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tam…

We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which …

Complete classification of knotoids up to seven crossings.

The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the generators of the fundamental group and their inverses: it is less than or equal to (L…

Paper adapts Bayesian Hui-Walter method for unlabeled data.

We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is ageneralization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three, …

The yearly aggregated tax income data of all, more than 8000, Italian municipalities are analyzed for a period of five years, from 2007 to 2011, to search for conformity or not with Benford's law, a counter-intuitive phenomenon observed in large tabulated data where the occurrence of numbers having smaller initial digi…

Machine learning classifies complex geometric patterns with high accuracy.

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in var…

This brief report (6 pages) was written in 1983 but never published. It concerns the hyperbolic 3-orbifolds obtained as quotients of hyperbolic 3-space by the group of invertible 2 by 2 matrices whose entries are integers in the imaginary quadratic extension of Q of discriminant D. For values D > -100 the topological t…

In this paper, we consider a framework adapting the notion of cointegration when two asset prices are generated by a driftless Itô-semimartingale featuring jumps with infinite activity, observed regularly and synchronously at high frequency. We develop a regression based estimation of the cointegrated relations method …

We give a new consistent scoring function for structure learning of Bayesian networks. In contrast to traditional approaches to score-based structure learning, such as BDeu or MDL, the complexity penalty that we propose is data-dependent and is given by the probability that a conditional independence test correctly sho…

Factor graphs have recently gained increasing attention as a unified framework for representing and constructing algorithms for signal processing, estimation, and control. One capability that does not seem to be well explored within the factor graph tool kit is the ability to handle deterministic nonlinear transformati…

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and struc…

A combinatorial framework classifies genus-one knots and links.

In an earlier paper we developed the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetri…

Motivation: Thanks to digitization, we often have access to large databases, consisting of various fields of information, ranging from numbers to texts and even boolean values. Such databases lend themselves especially well to machine learning, classification and big data analysis tasks. We are able to train classifier…

Algorithm finds mosaic numbers for knots with 10 or fewer crossings.