Models of random knots help understand typical knot behavior.
problem Understanding typical knot behavior from a probabilistic viewpoint.
method Presented several randomized models of knots and links, reviewed known results, discussed properties, and explored finite type invariants.
result Asymptotic distribution of knot invariants in random knots studied.
Study reveals a universal formula for knotting in random equilateral polygons.
problem Probability of knotting in equilateral random polygons.
method Extensive Monte Carlo simulations with improved algorithms and knot invariants.
result A universal scaling formula for knotting probability with number of edges, involving exponential and power law factors.
Study on crossing numbers of random two-bridge knots.
problem Understanding the distribution of crossing numbers in random two-bridge knots.
method Used billiard table diagrams to model random knots and derived a closed formula for their crossing numbers.
result Closed formula for the distribution of crossing numbers and exponential decay of knot appearance probability.
The article studies knot distributions in petal diagrams and proves probabilities of specific knot types decay as the number of petals increases.
problem Understanding the probability of specific knot types in petal diagrams as the number of petals grows.
method Established properties of the randomized knot model, proving probabilities decay to zero, improved bounds on crossing number and petal number relationships.
result The n-petal model represents at least exponentially many distinct knots, with probabilities of specific knot types decaying as the number of petals increases.
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.
Study on random knots and their projections with varying decay rates of coefficients.
problem Understanding the probability of knot types with a given number of crossings.
method Defined random knots as random periodic functions with Gaussian coefficients, analyzed decay rates of coefficients, and examined projections.
result For certain decay rates of coefficients, the probability of forming a knot type with a given number of crossings decays at least as fast as \(1/N\).
Study on writhe of permutations and random knots, with non-Gaussian distribution.
problem Understanding the writhe of permutations and its relation to random knots.
method Introduced writhe of permutations, studied asymptotics of random permutations, described model for random framed knots.
result Obtained a non-Gaussian limit distribution for the writhe of random permutations.
Study on random knot diagrams, proving unknots are rare and asymmetrical.
problem Understanding the probability and structure of random knot diagrams.
method Examined knot diagrams as topological maps, proving asymptotic laws.
result Unknot diagrams are asymptotically exponentially rare.
Study random knots and links via projections.
problem Model and analyze random knots and links through projections.
method Randomly project space curves onto 3D subspaces and analyze curvature and linking number.
result Compute the second moment of the linking number for random links.
New bounds on stick number of knots found using random polygon generation.
problem Understanding the minimum number of segments needed to build a polygonal knot.
method Monte Carlo approach to generating and analyzing large ensembles of random polygons.
result Improved bounds on stick number for over 40% of knots with 10 or fewer crossings.
Study on probabilities of knots in randomly generated diagrams.
problem Understanding the likelihood of different knot types in random diagrams.
method Generated random knot diagrams, classified by knot type, computed probabilities.
result Most diagrams are unknots, with a linear relationship between probability and frequency rank.
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…
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…
Random neural networks produce functions with a number of knots equal to the number of neurons.
problem Understanding why neural networks with many parameters do not overfit early in training.
method Analyzing random scalar-input feed-forward rectified linear unit architectures, showing they are random linear splines.
result The number of knots in random neural networks is equal to the number of neurons, to very close approximation.
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.
Study on knotting in very long polymer chains, finding Poisson distribution for prime knot types.
problem Understanding knotting in very long polymer chains.
method Generated and analyzed 243−k polygons of size n=2k using tree data structure and pivot algorithm. Used new knot diagram simplification and invariant-free classification. result Number of prime summands of knot type K in a random n-gon is well described by a Poisson distribution. 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.
Most graphs are knotted as they grow larger.
problem Understanding the prevalence of knotting in random graphs as they increase in size.
method Four models for random graphs were analyzed to determine the probability of intrinsic knotting.
result The probability of a graph being intrinsically knotted approaches 1 as the number of vertices increases.
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 …
The paper studies random covers of torus knot complements and their statistical properties.
problem Understanding the statistical behavior of finite covers of torus knot complements.
method Asymptotic subgroup growth analysis and Benjamini-Schramm limit theorems.
result Determination of the linear growth rate of Betti numbers for random covers of torus knot complements.
Study of knot invariant growth for twisted knots.
problem Understanding the behavior of a knot invariant for families of knots.
method Analyzing the perturbed Alexander invariant for twisted knots.
result Coefficients of the invariant grow linearly as the number of twists increases.
Study reveals weak knotting in confined polymers, not dominated by any single knot type.
problem Characterizing knotting in open, confined polymers.
method Modeling open curves as virtual knots, comparing lattice walks and ideal chains in confined and unconfined conditions.
result Weak knotting is a common feature in confined polymers, not dominated by any single knot type.
Develops a method for random manifolds and submanifolds, focusing on 3-ball knots.
problem Understanding random submanifolds in triangulated manifolds.
method Coloring vertices of triangulated manifolds to generate random submanifolds.
result Probability of generating an unknot decays exponentially in 3-ball with 3 colors.
New method identifies knots in protein chains using virtual knots.
problem Identifying knots in open protein chains.
method Introducing virtual knots to analyze open curves without closure.
result Recovering and extending previous knotting results in proteins.
Algorithm generates thick equilateral knots with controlled thickness.
problem Sampling and analyzing geometric knots with thickness constraints.
method Ergodic algorithm based on random reflections and thickness non-decreasing moves.
result Algorithm is ergodic and faster than previous methods, revealing strong growth in radius of gyration with thickness.
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.
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.
A new knot invariant is fast, strong, topologically meaningful, and fun.
problem Computing and understanding knot invariants efficiently and comprehensively.
method Developed a pair of polynomial knot invariants Θ=(Δ,θ) that are fast, strong, and topologically meaningful.
result Θ is a powerful knot invariant with separation power greater than other known invariants.
Locates entanglement in curves using knot intensity distribution.
problem Finding robust methods for locating entanglement in embedded curves.
method Introducing knot intensity distribution as a local quantifier for entanglement contribution.
result Intensity distributions identify regions in knots accommodating topological changes.
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.
Efficiently estimates covariance for sparse functional data.
problem Sparse data in functional analysis.
method Random-knots and B-spline estimators for covariance function.
result Asymptotic pointwise covariance estimates for sparsified data.
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.
Tensor networks speed up Jones polynomial calculation.
problem Efficiently calculating Jones polynomial for complex knots.
method Tensor network contraction for Potts model partition function.
result Jones polynomial can be evaluated subexponentially in knot complexity.
Given a knot K in an Euclidean space E and a finite dimensional space V of smooth functions on K, we express the expected number of critical points of a random function in V in terms of an integral-geometric invariant of K and V. When V consists of the restrictions to K of homogeneous polynomials of degree d on E, this…
It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones fun…
Method optimizes knotting pathways in constrained polymers.
problem Understanding how geometric constraints affect knot formation in polymers.
method Topological steering using knotoid spectrum and mean unravelling number.
result Geometric constraints increase the frequency of twist knots in polymers.
The UNKNOT problem solved using natural language processing and machine learning.
problem Determining if a knot is the unknot.
method Braid word representation, binary classification, Reformer and shared-QK Transformer networks, reinforcement learning, Markov moves, braid relations.
result Reformer and shared-QK Transformer networks outperform fully-connected networks in predicting the unknot.
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.
The paper computes special values of combinatorial zeta functions to reveal topological properties of manifolds.
problem Counting geodesic paths in triangulations to infer topological invariants.
method Random walks on higher-dimensional skeletons of triangulations.
result Recovery of Betti numbers and linking numbers of manifolds.
The study of random positive 3-strand braids reveals patterns in the roots of their Alexander polynomials.
problem Investigating the roots of Alexander polynomials of random positive 3-strand braids.
method Experimental data analysis, conjectures refinement, and proof of results using tools like the signature function of links and Lyapunov exponent of the Burau representation.
result Generically, at least 69% of the roots of Alexander polynomials are on the unit circle, with a large root-free region near the origin.
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.
Random walks on braid groups are transient, with specific closure properties for certain braids.
problem Understanding the behavior of random walks on braid groups and their closure properties.
method Analyzing the symplectic representation of braid groups and polynomial conditions on their matrices.
result Random walks on braid groups are transient, and specific closure properties for certain braids are derived.
Improved spatial prediction for massive datasets using SME model.
problem Efficiently estimating parameters in massive spatial datasets.
method Spatial Mixed Effects (SME) model with AECM algorithm for flexibility.
result Improved estimation without sacrificing prediction accuracy.
New invariant for square-free integers derived from kei theory.
problem Developing numerical invariants for square-free integers.
method Defining a kei for each square-free integer and calculating a coloring invariant.
result Conjecture and proof of asymptotic average order for coloring invariant.
EuLearn creates diverse 3D topological datasets for machine learning.
problem Training machine learning systems to discern topological features.
method Developed novel sampling and neural network architectures for graph and manifold data.
result Incorporating topological information improves deep learning performance on EuLearn datasets.
Polynomially parametrize interesting knotted surfaces.
problem Constructing polynomial parametrizations of knotted surfaces.
method Develop polynomial parametrization methods for specific knotted surfaces.
result Examples of polynomial parametrizations for knotted spheres, tori, and planes.
New 2-knots found with same knot group but different quandles.
problem Identifying 2-knots with identical knot groups but distinct quandles.
method Analyzing knot quandles of twist spins.
result First example of 2-knots with same knot group but different quandles.
Knots formed from torus knots are not concordant to L-space knots.
problem Understanding concordance in knots formed from connected sums of torus knots.
method Analyzing properties of knots formed from connected sums of torus knots.
result Knots formed from torus knots are not concordant to L-space knots.