Slipknots found in random diagrams almost always.
problem The presence of slipknots in random diagrams.
method Developed knotoid diagrams to study slipknots in knot diagrams.
result Almost all knot diagrams are slipknotted.
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48 results for “random diagrams”
We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random alternating diagrams, the expected hyperbolic volum…
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…
Study improves isoperimetric inequality for random groups.
problem Improving isoperimetric inequality for random groups.
method Generalizing the inequality to non-planar diagrams.
result Non-planar isoperimetric inequality established.
We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and pulling them uniformly from among sets of a given number of vertices , as first established in recent work with Cantarella and Mastin. The knot diagram model is an exciting new model which captures b…
Combinatorial Random Knotsmath.GT
Study on random knot diagrams and their probability of forming specific knots.
problem Understanding the probability of forming specific knots from random knot diagrams.
method Analyzing free knot diagrams without over/under information and proving trefoil formation; making conjectures about unknot and trefoil probabilities.
result Every free knot diagram produces trefoil knots, and certain families of diagrams are completely worked out.
Generates random persistence diagrams for data analysis.
problem Generating random persistence diagrams for data analysis.
method Based on pairwise interacting point processes and RJ-MCMC algorithm.
result Demonstrates the efficacy and utility of RPDG in materials science.
New model for links uses meander diagrams and combinatorics.
problem Modeling and analyzing random links.
method Random meander model based on meander diagrams and graphs, proving properties using combinatorics.
result Trivial links are unlikely, and there's a lower bound on non-isotopic knots.
We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this c…
We optimize large Random Forests into faster, smaller decision diagrams.
problem Efficiency and size of large Random Forests.
method Aggregating large Random Forests into a single, semantically equivalent decision diagram.
result Significant speed-ups and reduction in data structure size.
Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnosticstat.ML
Persistence diagrams from random matrices follow RMT universality, offering a new spectral diagnostic.
problem Understanding spectral properties of random matrices using topological data analysis.
method Applying Morse theory to persistence diagrams of quadratic forms restricted to unit spheres.
result Persistence entropy outperforms traditional level spacing ratios in discriminating random matrix ensembles.
Analogous zeta function for twisted Alexander invariants defined.
problem Defining a zeta function for twisted Alexander invariants.
method Modeling random walks on knot diagrams and interpreting Alexander polynomials and Jones polynomials as zeta functions.
result Analogous zeta function expression for twisted Alexander invariants.
Lower bounds on average genus of 2-bridge knots found.
problem Finding a lower bound on the average genus of 2-bridge knots.
method Developed a random model of 2-bridge knots, counted Seifert circles, and computed a lower bound.
result Computed a lower bound for the average Seifert genus of 2-bridge knots.
Develops path integral for spiked tensor model dynamics.
problem Dynamics of spiked tensor model with random initial conditions.
method Path integral approach applied to partial differential equations.
result Large- saddle point equations dominated by melonic diagrams.
To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus o…
The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of obtaining every specific knot type decays to…
Partial covariance factorizes in path diagrams, simplifying analysis.
problem Understanding partial covariance in complex diagrams.
method Factorization of partial covariance over nodes and edges.
result Simpson's paradox cannot occur in singly-connected diagrams.
Estimates and quantizes expected persistence diagrams for efficient analysis.
problem Statistical summary of the topology of structured data.
method Expected Persistence Diagram (EPD) and its quantization.
result Optimal estimation of EPD with near-optimal quantization.
Sharp density threshold for Property (T) found in random groups models.
problem Density threshold for Kazhdan's Property (T) in random groups.
method Quotient of free groups by random reduced words, new geometrical tools.
result Sharp density threshold for Property (T) equals 1/3.
The paper introduces a new Markov chain sampler for knot diagrams.
problem Efficiency of existing sampling methods for knot diagrams is limited.
method Local moves based on Reidemeister moves to sample plane curves, then map to knot diagrams.
result Achieved an efficient sampler of knot diagrams and analyzed their asymptotic behavior.
Introduces a new space of Radon measures for better understanding persistence diagrams.
problem Lack of optimal transport-based formalism for persistence diagrams.
method Formalizes persistence diagrams as Radon measures on the upper half plane via optimal partial transport.
result Characterizes convergence and barycenters of persistence diagrams.
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a …
Improved modeling of persistence diagrams for data analysis.
problem Determining significant outliers in persistence diagrams.
method Modification of the RST (Replicating Statistical Topology) model using MCMC Metropolis-Hastings algorithm.
result The modified RST model improves the goodness of fit in persistence diagram analysis.
Study on typical knots and links using grid diagrams, focusing on size, components, and writhe.
problem Understanding the statistical behavior of knots and links, especially their typical properties.
method Modeling knots and links with grid diagrams, examining three invariants: size, components, and writhe, through numerical analysis.
result The size of a random knot is uniformly distributed and linearly dependent on grid size, while the number of components follows a distribution whose mean and variance grow with log_2 of grid size.
KnotMosaics package simplifies knot theory computations in SageMath.
problem Efficiently computing knot mosaic diagrams and their properties.
method Developed a SageMath package for knot mosaic diagrams, implementing validation, strand tracing, and computation algorithms.
result Enabled easy computation of knot mosaic diagrams and their properties.
Our main result is that for densities a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality, to some class of non-planar diagrams and, using this, we introduce a system of modified hypergraphs providing the structure o…
In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to ze…
Study of two-layer ReLU neural network phase diagram at infinite-width limit.
problem Characterize the dynamical regimes of two-layer ReLU neural networks.
method Combining experimental and theoretical approaches, including phase diagram analogy.
result Identification of three regimes: linear, critical, and condensed.
We describe a bottom-up framework, based on the identification of appropriate order parameters and determination of phase diagrams, for understanding progressively refined agent-based models and simulations of financial markets. We illustrate this framework by starting with a deterministic toy model, whereby indepe…
We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of t…
A new method uses vectorized summaries of persistence diagrams for efficient hypothesis testing.
problem Efficient hypothesis testing for large and complex persistence diagrams.
method Vectorized summaries of Betti functions and a new shuffling technique.
result The vectorized Betti function leads to competitive results compared to baseline methods.
Random groups prove length constraints on product of conjugates.
problem Quantify products of conjugates in random groups.
method Sharp van Kampen diagram argument and boundary block-counting.
result Prove a sharp inequality for products of conjugates in random groups.
ReAPR simplifies hard unknots by reembedding and rerouting, revealing hidden simplifications.
problem Training AI to recognize knots, especially hard unknots, is challenging.
method Alternates pass-move reduction with geometric re-embedding, minimizing total variation of a height function.
result ReAPR successfully simplifies hard unknots, including Kauffman's challenge unknots, in under 30 seconds.
New spectral Dehn function characterizes word-hyperbolic groups.
problem Characterizing word-hyperbolic groups using spectral properties.
method Introducing spectral Dehn functions and proving inequalities relating them to Dehn functions.
result A spectral Dehn function characterizes word-hyperbolic groups.
We introduce a new random group model called the square model: we quotient a free group on generators by a random set of relations, each of which is a reduced word of length four. We prove, as in the Gromov density model, that for densities a random group in the square model is trivial with overwhel…
New method recovers graph latent positions under edge differential privacy.
problem Recovering latent graph information from privatized graphs.
method Applying geometric insights to adjust statistical inference for privatized graphs.
result Achieves consistent recovery of latent positions under local edge differential privacy constraints.
Let be the genus of a two-dimensional surface obtained by gluing, uniformly at random, the sides of an -gon. Recently Linial and Nowik proved, via an enumerational formula due to Harer and Zagier, that the expected value of is asymptotic to for . We prove a local limit theorem…
Study recovers neuron assemblies from cognitive data using tensor decomposition.
problem Recovering neuron assemblies from cognitive data.
method Linear independence analysis of rank one tensors, polynomial measurements.
result Reconstructs Venn diagram of neuron assemblies from intersection sizes.
The paper examines linking numbers in grid models and finds polynomial moments.
problem Analyzing linking numbers in grid models.
method Examined linking numbers as a random variable on isotopy classes of 2-component links, computed moments and limits.
result The th moment of the linking number is a polynomial in the grid size with degree , and all odd moments vanish.
The paper refines transformations of lattice diagrams and introduces dotted diagrams.
problem Investigating transformations and deformations of lattice diagrams and their associated dotted diagrams.
method Introducing dotted diagrams and investigating deformations of these diagrams, relating them to transformations of lattice diagrams.
result Refined results on the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams.
Estimates BV functions from noisy data using Voronoi diagrams.
problem Estimating multivariate BV functions from scattered noisy data.
method Form Voronoi diagram, solve optimization problem with discrete TV regularization.
result Voronoigram is minimax rate optimal for BV functions.
This note explains how to transform Heegaard diagrams into framed link diagrams.
problem No specific problem stated; transformation of diagrams is the focus.
method Explains a procedure to transform Heegaard diagrams into framed link diagrams.
result Demonstrates a method to transform Heegaard diagrams into framed link diagrams.
New minimal link diagrams found, including torus links and homogeneous ones.
problem Finding minimal link diagrams with new classes.
method Morton-Franks-Williams inequality approach.
result New classes of minimal link diagrams, including previously unproven ones.
Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.
problem Creating efficient trisection diagrams for 4-manifolds.
method Algorithm converting Kirby diagrams to trisection diagrams.
result Provides examples of trisection diagrams for 4-manifolds.
Kernelized Taylor diagram visualizes data populations with fewer assumptions.
problem Limitations of Taylor diagram in capturing non-linear relationships and sensitivity to outliers.
method Proposes a kernelized version of the Taylor diagram that uses maximum mean discrepancy and kernel mean embedding.
result Kernelized Taylor diagram visualizes data populations with minimal assumptions of data distributions.
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram by use of the double covering …
Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower bound for the virtual crossing number of virtual graph diagrams.