Set Of Knots Research Articles

A framework for generalizing toric inequalities for holographic entanglement entropy

We conjecture a multi-parameter generalization of the toric inequalities of [1]. We then extend their proof methods for the generalized toric inequalities in two ways. The first extension constructs the graph corresponding to the toric inequalities and the generalized toric conjectures by tiling the Euclidean space. An entanglement wedge nesting relation then determines the geometric structure of the tiles. In the second extension, we exploit the cyclic nature of the inequalities and conjectures to construct cycle graphs. Then, the graph can be obtained using graph Cartesian products of cycle graphs. In addition, we define a set of knots on the graph by following [1]. These graphs with knots then imply the validity of their associated inequality. We study the case where the graph can be decomposed into disjoint unions of torii. Under the specific case, we explore and prove the conjectures for some ranges of parameters. We also discuss ways to explore the conjectured inequalities whose corresponding geometries are d-dimensional torii (d > 2).

Large-Scale Low-Rank Gaussian Process Prediction with Support Points

Low-rank approximation is a popular strategy to tackle the “big n problem” associated with large-scale Gaussian process regressions. Basis functions for developing low-rank structures are crucial and should be carefully specified. Predictive processes simplify the problem by inducing basis functions with a covariance function and a set of knots. The existing literature suggests certain practical implementations of knot selection and covariance estimation; however, theoretical foundations explaining the influence of these two factors on predictive processes are lacking. In this paper, the asymptotic prediction performance of the predictive process and Gaussian process predictions are derived and the impacts of the selected knots and estimated covariance are studied. The use of support points as knots, which best represent data locations, is advocated. Extensive simulation studies demonstrate the superiority of support points and verify our theoretical results. Real data of precipitation and ozone are used as examples, and the efficiency of our method over other widely used low-rank approximation methods is verified.

Homomorphism obstructions for satellite maps

A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in S 3 S^3 . This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides ± 1 \pm 1 . We then provide subtle examples of satellite maps which map arbitrarily deep into the n n -solvable filtration of Cochran, Orr, and Teichner [Ann. of Math. (2) 157 (2003), pp. 433–519], act like homomorphisms on arbitrary finite sets of knots, and yet which still do not induce homomorphisms. Finally, we verify Hedden’s conjecture in the smooth category for all small crossing number satellite operators but one.

A weighted online regularization for a fully nonparametric model with heteroscedasticity

<abstract><p>In this paper, combining B-spline function and Tikhonov regularization, we propose an online identification approach for reconstructing a smooth function and its derivative from scattered data with heteroscedasticity. Our methodology offers the unique advantage of enabling real-time updates based on new input data, eliminating the reliance on historical information. First, to address the challenge of heteroscedasticity and computation cost, we employ weight coefficients along with a judiciously chosen set of knots for interpolation. Second, a reasonable approach is provided to select weight coefficients and the regularization parameter in objective functional. Finally, We substantiate the efficacy of our approach through a numerical example and demonstrate its applicability in solving inverse problems. It is worth mentioning that the algorithm not only ensures the calculation efficiency, but also trades the data accuracy through the data volume.</p></abstract>

Ribbon concordance of knots is a partial ordering

In this note we show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon [Math. Ann. 257 (1981), pp. 157–170, Conjecture 1.1]. The proof makes use of representation varieties of the knot groups to S O ( N ) SO(N) and the subquotient relation between them induced by a ribbon concordance.

Asymptotic behavior of an intrinsic rank-based estimator of the Pickands dependence function constructed from B-splines

A bivariate extreme-value copula is characterized by its Pickands dependence function, i.e., a convex function defined on the unit interval satisfying boundary conditions. This paper investigates the large-sample behavior of a nonparametric estimator of this function due to Cormier et al. (Extremes 17:633–659, 2014). These authors showed how to construct this estimator through constrained quadratic median B-spline smoothing of pairs of pseudo-observations derived from a random sample. Their estimator is shown here to exist whatever the order m ge 3 of the B-spline basis, and its consistency is established under minimal conditions. The large-sample distribution of this estimator is also determined under the additional assumption that the underlying Pickands dependence function is a B-spline of given order with a known set of knots.

A process convolution model for crash count data on a network

Crash data observed on a road network often exhibit spatial correlation due to unobserved effects with inherent spatial correlation following the structure of the road network. It is important to model this spatial correlation while accounting for the road network structure. In this study, we introduce the network process convolution (NPC) model. In this model, the spatial correlation among crash data is captured by a Gaussian Process (GP) approximated through a kernel convolution approach. The GP’s covariance function is based on path distance computed between a limited set of knots and crash data points on the road network. The proposed model offers a straightforward approach for predicting crash frequency at unobserved locations where covariates are available, and for interpolating the GP values anywhere on the network. Inference procedure is performed following the Bayesian paradigm and is implemented in R-INLA, which offers an estimation procedure that is very efficient compared to Markov Chain Monte Carlo sampling algorithms. We fitted our model to synthetic data and to crash data from Ottawa, Canada. We compared the proposed approach with a proper Conditional Autoregressive (pCAR) model, and with Poisson Regression (PR) and Negative Binomial (NB) models without latent effects. The results of the study indicated that although the pCAR model has comparable fitting performance, the NPC model outperforms pCAR when the main goal is to predict unobserved locations of interest. The proposed model also offers lower mean absolute error rates for cross validated crash counts, latent variable values, fixed-effect coefficients, as well as shorter interval scores for singletons. The NPC provides a natural way to account for the road network structure when considering the inclusion of spatially structured latent random effects in the modelling of crash data. It also offers an improved predictive capability for crash data on a road network.

Homology concordance and an infinite rank free subgroup

Two knots are homology concordant if they are smoothly concordant in a homology cobordism. The group C ̂ Z (respectively, C Z ) was previously defined as the set of knots in homology spheres that bounds homology balls (respectively, in S 3 ), modulo homology concordance. We prove C ̂ Z / C Z contains a Z ∞ subgroup. We construct our family of examples by applying the filtered mapping cone formula to L-space knots, and prove linear independence with the help of the connected knot complex.

Simple Smale flows and their templates on S3

The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of 3-dimensional flows. We prove that for an embedded template in S3 with fixed homeomorphism type, its boundary as a trivalent spatial graph is a complete isotopic invariant. Moreover, we construct an invariant of embedded templates by Kauffman's invariant of spatial graphs, which is a set of knots and links. As an application, the isotopic classification of simple Smale flows on S3 is discussed.

A Bayesian Spatial Categorical Model for Prediction to Overlapping Geographical Areas in Sample Surveys

Summary Motivated by the Australian National University poll, we consider a situation where survey data have been collected from respondents for several categorical variables and a primary geographic classification, e.g. postcode. Here, a common and important problem is to obtain estimates for a second target geography that overlaps with the primary geography but has not been collected from the respondents. We examine this problem when areal level census information is available for both geographic classifications. Such a situation is challenging from a small area estimation perspective for several reasons: there is a misalignment between the census and survey information as well as the geographical classifications; the geographic areas are potentially small and so prediction can be difficult because of the sparse or spatially missing data issue; and there is the possibility of non-stationary spatial dependence. To address these problems we develop a Bayesian model using latent processes, underpinned by a non-stationary spatial basis that combines Moran's I and multiresolution basis functions with a small but representative set of knots. The study results based on simulated data demonstrate that the model can be highly effective and gives more accurate estimates for areas defined by the target geography than several existing models. The model also performs well for the Australian National University poll data to predict on a second geographic classification: statistical area level 2.

Knot Floer homology obstructs ribbon concordance

We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. Generalizing theorems of Gabai and Scharlemann, we also prove that the Seifert genus is super-additive under band connected sums of arbitrarily many knots. Our results give evidence for a conjecture of Gordon that ribbon concordance is a partial order on the set of knots.

A finite element framework based on bivariate simplex splines on triangle configurations

Recently, triangle configuration based bivariate simplex splines (referred to as TCB-spline) have been introduced to the geometric computing community. TCB-splines retain many attractive theoretic properties of classical B-splines, such as partition of unity, local support, polynomial reproduction and automatic inbuilt high-order smoothness. In this paper, we propose a computational framework for isogeometric analysis using TCB-splines. The centroidal Voronoi tessellation method is used to generate a set of knots that are distributed evenly over the domain. Then, knot subsets are carefully selected by a so-called link triangulation procedure (LTP), on which shape functions are defined in a recursive manner. To achieve high-precision numerical integration, triangle faces served as background integration cells are obtained by triangulating the entire domain restricted to all knot lines, i.e., line segments defined by any two knots in a knot subset. Various numerical examples are carried out to demonstrate the efficiency, flexibility and optimal convergence rates of the proposed method.

The distribution of knots in the Petaluma model

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 zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots. Past approaches to showing, in some random models, that individual knot types occur with vanishing probability, rely on the prevalence of localized connect summands as the complexity of the knot increases. However this phenomenon is not clear in other models, including petal diagrams, random grid diagrams, and uniform random polygons. Thus we provide a new approach to investigate this question.

The H(n)-Gordian complex of knots

In this paper, the [Formula: see text]-Gordian complex of knots is defined. The authors show that for any knot [Formula: see text] and any positive integer [Formula: see text], there is a finite set of knots of size [Formula: see text] containing [Formula: see text], such that the [Formula: see text]-Gordian distance between any two knots in this set is 1 for all [Formula: see text].

Learning to tie the knot: The acquisition of functional object representations by physical and observational experience.

Here we examined neural substrates for physically and observationally learning to construct novel objects, and characterized brain regions associated with each kind of learning using fMRI. Each participant was assigned a training partner, and for five consecutive days practiced tying one group of knots (“tied” condition) or watched their partner tie different knots (“watched” condition) while a third set of knots remained untrained. Functional MRI was obtained prior to and immediately following the week of training while participants performed a visual knot-matching task. After training, a portion of left superior parietal lobule demonstrated a training by scan session interaction. This means this parietal region responded selectively to knots that participants had physically learned to tie in the post-training scan session but not the pre-training scan session. A conjunction analysis on the post-training scan data showed right intraparietal sulcus and right dorsal premotor cortex to respond when viewing images of knots from the tied and watched conditions compared to knots that were untrained during the post-training scan session. This suggests that these brain areas track both physical and observational learning. Together, the data provide preliminary evidence of engagement of brain regions associated with hand-object interactions when viewing objects associated with physical experience, and with observational experience without concurrent physical practice.

Modified spline regression based on randomly right-censored data: A comparative study

ABSTRACTIn this paper, we propose modified spline estimators for nonparametric regression models with right-censored data, especially when the censored response observations are converted to synthetic data. Efficient implementation of these estimators depends on the set of knot points and an appropriate smoothing parameter. We use three algorithms, the default selection method (DSM), myopic algorithm (MA), and full search algorithm (FSA), to select the optimum set of knots in a penalized spline method based on a smoothing parameter, which is chosen based on different criteria, including the improved version of the Akaike information criterion (AICc), generalized cross validation (GCV), restricted maximum likelihood (REML), and Bayesian information criterion (BIC). We also consider the smoothing spline (SS), which uses all the data points as knots. The main goal of this study is to compare the performance of the algorithm and criteria combinations in the suggested penalized spline fits under censored data. A Monte Carlo simulation study is performed and a real data example is presented to illustrate the ideas in the paper. The results confirm that the FSA slightly outperforms the other methods, especially for high censoring levels.

Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Suppose C is a singular curve in \mathbb CP^2 and it is topologically an embedded surface of genus g ; such curves are called cuspidal. The singularities of C are cones on knots K_i . We apply Heegaard Floer theory to find new constraints on the sets of knots \{K_i\} that can arise as the links of singularities of cuspidal curves. We combine algebro-geometric constraints with ours to solve the existence problem for curves with genus one, d > 33 , that possess exactly one singularity which has exactly one Puiseux pair (p;q) . The realized triples (p,d,q) are expressed as successive even terms in the Fibonacci sequence.

Finite type invariants of w-knotted objects II: tangles, foams and the Kashiwara–Vergne problem

This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are wider but weaker than their “usual” counterparts. To get (say) w-knots from usual knots (or u-knots), one has to allow non-planar “virtual” knot diagrams, hence enlarging the base set of knots. But then one imposes a new relation beyond the ordinary collection of Reidemeister moves, called the “overcrossings commute” relation, making w-knotted objects a bit weaker once again. Satoh (J. Knot Theory Ramif. 9-4:531–542, 2000) studied several classes of w-knotted objects (under the name “weakly-virtual”) and has shown them to be closely related to certain classes of knotted surfaces in $${\mathbb R}^4$$ . In this article we study finite type invariants of w-tangles and w-trivalent graphs (also referred to as w-tangled foams). Much as the spaces $${\mathcal A}$$ of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces $${\mathcal A}^w$$ of “arrow diagrams” for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of w-foams is essentially the same as a solution of the Kashiwara and Vergne (Invent. Math. 47:249–272, 1978) conjecture and much of the Alekseev and Torossian (Ann. Math. 175:415–463, 2012) work on Drinfel’d associators and Kashiwara–Vergne can be re-interpreted as a study of w-foams.

Satellite operators as group actions on knot concordance

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\overline{P}$ for which $\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in a potentially exotic $S^3\times [0,1]$) for all knots K; we show that these satellite operators are distinct from all connected-sum operators, even up to concordance, and that they induce bijective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture.

Knot homotopy in subspaces of the 3-sphere

We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere $S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we show that every knot in a subspace of the 3-sphere is transient if and only if the exterior of the subspace is a disjoint union of handlebodies, i.e. regular neighborhoods of embedded graphs, where a knot in a 3-subspace of $S^3$ is said to be transient if it can be moved by a homotopy within the subspace to the trivial knot in $S^3$. To show this, we discuss relation between certain group-theoretic and homotopic properties of knots in a compact 3-manifold, which can be of independent interest. Further, using the notion of transient knot, we define an integer-valued invariant of knots in $S^3$ that we call the transient number. We then show that the union of the sets of knots of unknotting number one and tunnel number one is a proper subset of the set of knots of transient number one.