A new knot selection method speeds up sparse Gaussian process approximations.
problem Efficiently selecting knots for sparse Gaussian processes.
method One-at-a-time Bayesian optimization for knot selection.
result Competitive performance with reduced computational cost.
Proposes a more efficient knot selection method for sparse Gaussian processes.
problem Optimizing marginal likelihood for knot selection leads to suboptimal and inefficient placement of knots.
method Uses Bayesian optimization to propose knots one at a time, avoiding multimodal surface issues.
result Improves both accuracy and speed of knot selection compared to current methods.
Generalizes region select game to k-colored knot diagrams.
problem Play game on knot diagrams with multiple colors.
method Generalize region select game to k-colored knot diagrams. result Generalization of the region select game to k-colored knot diagrams. A new method optimizes knot selection for spline dimensional decomposition in stochastic dynamic analysis.
problem Challenges in uncertainty quantification for dynamical systems with non-smooth or oscillating nonlinear behaviors.
method Interpolation-based optimal knot selection method for SDD, improving accuracy and computational efficiency.
result SDD with proposed knot selection yields higher accuracy than other methods, as shown in a lower control arm example.
New game defined on origami patterns, linking number introduced.
problem Defining a game on origami patterns.
method Introduced Region Select on origami crease patterns.
result Defined a new unlinking number.
Deep P-Spline automates DNN structure selection for complex regression problems.
problem Challenges in selecting optimal network structures for DNNs.
method Linking neuron selection to knot placement in basis expansion techniques, introducing a difference penalty for automated knot selection.
result Deep P-Spline extends model class and forms a latent variable modeling framework with theoretical guarantees.
New moves for singular knots identified and described.
problem Identifying and describing moves for singular knots.
method Provided 96 generating sets of oriented singular Reidemeister moves and selected moves for Legendrian singular knots.
result Surviving moves for Legendrian singular knots were identified and described.
Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its origin…
A knot k is called ``strongly (n-1)-trivial.'' if there exists a projection of k, such that one can choose n crossings of the projection with the property that making the crossing changes corresponding to any of the 2n−1 nontrivial combinations of the selected crossings turns the original knot into the unknot. We …
Study inequalities between knot invariants and compute new bounds.
problem Understanding inequalities and bounds between knot invariants.
method Directed graph of inequalities, database construction, systematic transitivity criterion.
result 139 new exact values for unknotting number and doubly slice genus.
We investigate multiple testing and variable selection using the Least Angle Regression (LARS) algorithm in high dimensions under the assumption of Gaussian noise. LARS is known to produce a piecewise affine solution path with change points referred to as the knots of the LARS path. The key to our results is an express…
Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he re…
We present in this chapter (Chapter II) the history of ideas which lead up to the development of modern knot theory. We are more detailed when pre-XX century history is reported. With more recent times we are more selective, stressing developments related to Jones type invariants of links. In the Appendix, A.Przybyszew…
New method detects and compares folding pathways of knotted proteins.
problem Understanding the function of knots in protein folding.
method Topological analysis of protein knotoid distributions and entanglement.
result Reveals unique folding pathway for shallow knotted Carbonic Anhydrases.
We address the issue of knots selection for Gaussian predictive process methodology. Predictive process approximation provides an effective solution to the cubic order computational complexity of Gaussian process models. This approximation crucially depends on a set of points, called knots, at which the original proces…
This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossing…
The image of a polygonal knot K under a spherical inversion of R^3 (union infinity) is a simple closed curve made of arcs of circles, having the same knot type as the mirror image of K. Suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon. This may be a different knot type…
This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (T …
A machine learning method selects optimal orthonormal bases for functional data analysis.
problem Lack of formal criteria for choosing initial orthonormal bases in functional data methods.
method Proposes a machine learning algorithm to learn and place knots for efficient orthogonal spline bases (splinets).
result Demonstrates efficiency, especially for sparse functional data and complex physical systems.
Authors prove a contact structure result using branched covers and overtwisted disks.
problem Proving a contact structure result using branched covers and overtwisted disks.
method Explicitly constructing an overtwisted disk in the p-fold cyclic branched cover of S3. result An overtwisted disk is contained in the complement of the branch locus.
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.
We extend the adaptive regression spline model by incorporating saturation, the natural requirement that a function extend as a constant outside a certain range. We fit saturating splines to data using a convex optimization problem over a space of measures, which we solve using an efficient algorithm based on the condi…
The paper classifies palettes of Dehn colorings for spatial graphs.
problem Classifying spatial graph diagrams using Dehn colorings.
method Examining vertex conditions and palettes for spatial graphs.
result Spatial graphs can be distinguished by the number of Dehn colorings with specific palettes.
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.
New knot quandles distinguish ribbon knots with isomorphic groups.
problem Distinguishing knots with isomorphic fundamental groups.
method Examined knot quandles of Suciu's ribbon knots and computed their types.
result Knot quandles of Suciu's ribbon knots are mutually non-isomorphic.
Proved colored HOMFLY-PT polynomials for specific knots.
problem Calculating colored HOMFLY-PT polynomials for specific knots.
method Rigorous mathematical proof for trefoil, figure-eight, and twist knots.
result Colored HOMFLY-PT polynomials expressed as sums for different knots.
Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …
The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.
problem Proving the non-triviality of welded knots and ribbon torus-knots.
method By generating examples and determining the fundamental group of the concerned welded knot.
result Non-triviality of welded knots and ribbon torus-knots is demonstrated.
We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transver…
Study concordance of alternating torus knots to L-space knots.
problem When are linear combinations of alternating torus knots concordant to L-space knots?
method Proved Allen's conjecture for alternating torus knots and established a necessary condition.
result Linear combinations of alternating torus knots are concordant to L-space knots if and only if they are a single torus knot.
This paper studies how knots combine using Alexander Polynomials.
problem How knots combine and their determinants behave.
method Basic knot theory, Alexander Polynomials, and composition techniques.
result Generalized solution for knot determinants in compositions.
The paper classifies a special family of knots in lens spaces using knot Floer homology.
problem Classifying constrained knots in lens spaces.
method Parameterization by five integers, characterization via spinc structures, and knot Floer homology calculations. result Complete classification of constrained knots based on knot Floer homology.
The study confirms conjectures about slopes of knots using knot Floer homology.
problem Verifying conjectures about non-integer characterizing slopes of knots.
method Using knot Floer homology, the study verifies conjectures for specific classes of knots.
result Almost all slopes are characterizing for many knots, and infinitely many for L-space knots. Defines slice depth for 2-knots and sets upper bounds for specific knots.
problem Determining the minimum dimension for a 2-knot to be slice.
method Introduces slice depth, defines it for 2-knots, and provides upper bounds for specific knot types.
result Upper bounds for slice depth of certain 2-knots.
New diagonal knots found with non-torus structure.
problem Identifying knots with diagonal grid diagrams.
method Analysis of knots represented by diagonal grid diagrams.
result All diagonal knots are positive, and a new non-torus example is found.
New hyperbolic knots not concordant to algebraic ones found.
problem Identifying knots not concordant to algebraic knots.
method Constructing hyperbolic L-space knots.
result Found hyperbolic knots that are not concordant to algebraic knots.
Formula for Alexander polynomial of twisted torus knots derived.
problem Calculating Alexander polynomial for a specific class of knots.
method Knot group presentation combined with Fox's calculus.
result Explicit formula for Alexander polynomial of twisted torus knots.
A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecan…
Expanded Legendrian knot atlas for 10-arc index knots.
problem Lack of Legendrian knot data for knots with high arc index.
method Created an atlas of Legendrian knots up to arc index 10.
result Legendrian knots of arc index 10 have been cataloged.
The paper conjectures Khovanov homology can distinguish torus and twist knots.
problem Detecting and distinguishing knots using Khovanov homology.
method Examining all prime knots with up to 20 crossings, conjecturing Legendrian simplicity.
result Numerical evidence supports Khovanov homology distinguishing torus and twist knots.
Two complete knot invariants from diagrams, finite or infinite.
problem Classifying knots completely.
method Constructed two invariants from knot diagrams, finite or infinite.
result Finite set reveals knotting number.
Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.
problem Determine grid homology of diagonal knots and compare them to other knot types.
method Use grid diagrams and combinatorial knot Floer homology to analyze diagonal knots.
result Grid homology detects the number of prime factors and decompositions of the knot into non-integer tangles.
New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.
Polynomially parameterizes knots and spheres, proving analogous results.
problem Parameterizing knots and spheres using polynomials.
method Analogous to classical knots, parameterized long 2-knots and certain classes of knotted spheres.
result Polynomial parameterizations for knotted spheres constructed.
New infinite families of twisted torus knots found.
problem Identifying new types of twisted torus knots.
method Finding new infinite families of twisted torus knots with a single negative twist.
result Eight new infinite families of twisted torus knots are discovered.
New knot concept extends welded knots, simplifying classification.
problem Classifying welded knots and their complements.
method Introducing 'wen knots', proving subset relationships, characterizing complements.
result Extended welded knots can be fully characterized by the parity of wens.
We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number of cylinder knots with two fixed parameters (out of three). In addition we prove …