Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for weaving knots, and we prove that the infinite weave is their geometric limit.
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48 results for “weaving knots”
Study on bounds of knot untangling for specific types of knots.
problem Determining upper limits for knot untangling.
method Defined warping degree, examined diagrams combinatorially.
result Upper bounds for unknotting and region unknotting numbers.
Study links weaving knots with polynomial coefficients and lattice numbers.
problem Understanding polynomial coefficients of weaving knots and their lattice counterparts.
method Established relationships between Jones and Chebyshev polynomials, and derived explicit formulas for Alexander polynomials.
result Proved coefficients of Jones polynomial are Whitney numbers of Lucas lattices and satisfied Fox's trapezoidal conjecture.
Formula found for knot determinant in 3-braid weaving.
problem Determining the determinant of a specific type of knot.
method Developed a formula for the determinant of the twisted generalized hybrid weaving knot.
result Proved a conjecture about the knot's determinant.
The paper calculates properties of weaving knots and their unknotting numbers.
problem Calculating properties of specific weaving knots.
method Deriving formulae for knot determinants and homology groups.
result Lower bounds on unknotting numbers for certain weaving knots.
In this paper we compute the signature for a family of knots , the weaving knots of type . By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots , we develop recursion relations that enable us to compute the Jo…
Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In th…
We investigate several conjectures in geometric topology by assembling computer data obtained by studying weaving knots, a doubly infinite family of examples of hyperbolic knots. In particular, we compute some important polynomial knot invariants, as well as knot homologies, for the subclass of this f…
Paper characterizes isotopies and hyperbolicity of weaves using geodesics.
problem Characterizing isotopies and hyperbolicity of weaves.
method Using diagrams of closed geodesics and normal positions of essential surfaces.
result Weaves are hyperbolic and cannot have essential Conway spheres.
This paper classifies periodic weaves and their universal cover, extending Tait's conjectures.
problem Classifying periodic weaves and their universal cover in thickened surfaces.
method Introducing hyperbolic periodic weaves, extending Tait's conjectures, and using a generalized Kauffman bracket polynomial.
result Tait's conjectures are extended to minimal reduced alternating weaving motifs.
Algorithm constructs and classifies weaving diagrams using combinatorial methods.
problem Classifying unique weaving diagrams with over/under information.
method Systematic algorithm based on tiling and crossing matrices.
result Classification of periodic structures based on minimum crossings.
This paper classifies a specific weave type by their crossing number.
problem Classifying doubly periodic untwisted (p,q)-weaves.
method Classification by crossing number, introducing crossing matrix for equivalence.
result Classification of untwisted (p,q)-weaves by their crossing number.
Proves certain alternating links have specific geometric properties.
problem Characterizing alternating links with totally geodesic checkerboard surfaces.
method Analyzes links with two totally geodesic checkerboard surfaces and characterizes them.
result Proves links with two totally geodesic checkerboard surfaces are three specific links.
Weaved helices form mechanically stable 3D structures.
problem Creating stable 3D structures from helical elements.
method Exploiting screw symmetry and invariant cylindrical rod packing to form triply periodic arrangements.
result Demonstrated nineteen triply periodic arrangements of interwoven helices.
Paper constructs motifs from planar tilings for DP weaves and polycatenanes.
problem Creating complex entangled structures from periodic tilings.
method Combinatorial methodology using polygonal link transformations.
result Predicting the type of motif from a given tiling and polygonal link method.
Survey of Turk's head knots and links properties.
problem Characterize Turk's head knots and links.
method Discussion of various results in the mathematical literature.
result Turk's head links are alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime.
We prove the existence of Lagrangian fillings for -type Legendrian links.
problem Exact Lagrangian fillings of Legendrian links of -type.
method Legendrian weave calculus and construction of 1-cycles.
result Existence of a Lagrangian filling represented by a weave.
The paper connects different types of Lagrangian fillings to Legendrian weaves and their sheaf quantizations.
problem Understanding and comparing different types of Lagrangian fillings of Legendrian weaves.
method Establishing new Reidemeister moves and combinatorial isotopies between Lagrangian fillings, comparing sheaf quantizations.
result Legendrian weaves generalize previously known methods to produce infinitely many distinct Lagrangian fillings.
New method weaves paper strips for designing curved surfaces with elasticity.
problem Designing general curved surfaces with geometrical elasticity.
method Shape optimization of paper strips using nonlinear elasticity theory.
result Demonstrated creation of catenoid and helicoid surfaces with 54 paper strips.
Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose th term is the th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the -th colored Jones polynomial at $\exp(2 πi \a/n)$ is a sequence of complex numbers that grows exponentially, for a fixe…
In target tracking, the estimation of an unknown weaving target frequency is crucial for improving the miss distance. The estimation process is commonly carried out in a Kalman framework. The objective of this paper is to examine the potential of using neural networks in target tracking applications. To that end, we pr…
Study identifies stable configurations of intertwined threads with repulsive interactions.
problem Stable configurations of entangled systems with repulsive interactions.
method Analysis of steepest descent flow of an energy functional.
result Existence and uniqueness of stable configuration of two layers drifting apart at rate.
Paper defines untangling number to measure entanglement complexity in 3-periodic networks.
problem Measuring the complexity of entanglement in 3-periodic networks.
method Defining ground states through knot-theoretic crossing diagrams and measuring untangling number.
result Introduced untangling number as a measure of entanglement complexity.
Machine learning is often used in virtual screening to find compounds that are pharmacologically active on a target protein. The weave module is a type of graph convolutional deep neural network that uses not only features focusing on atoms alone (atom features) but also features focusing on atom pairs (pair features);…
Non-Reidemeister Knot Theory and Its Applications in Dynamical Systems, Geometry, and Topologymath.GT
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to diagrams of the initial object. The main tool is the notion of {\em free -braid …
Study detects a specific type of link using annular Khovanov homology.
problem Detecting a specific type of three-strand weaving link.
method Combines braid detection with rigidity theorem to determine up to conjugacy.
result Annular Khovanov homology detects the underlying unoriented annular link .
Generalizing previous constructions, we present a dual pair of decompositions of the complement of a link L into bipyramids, given any multi-crossing projection of L. When L is hyperbolic, this gives new upper bounds on the volume of L given its multi-crossing projection. These bounds are realized by three closely rela…
TempoPFN models for zero-shot time series forecasting using synthetic data.
problem Efficient long-horizon prediction and reproducibility in zero-shot time series forecasting.
method Linear RNNs pre-trained on synthetic data with GatedDeltaProduct architecture and state-weaving.
result Achieves top-tier competitive performance on various benchmarks.
New surfaces with special geodesic and horocycle behaviors discovered.
problem Understanding geodesic and horocycle dynamics on hyperbolic surfaces.
method Constructing geometrically infinite hyperbolic surfaces with tailored recurrence properties.
result First examples of non-trivial minimal horocyclic orbit closures and infinite locally-finite conservative horocyclic invariant measures.
Study Legendrian surfaces using N-graphs and flag moduli.
problem Characterize and apply Legendrian surfaces in contact geometry.
method Develop diagrammatic calculus and algebraic-geometric characterization.
result Show applications in Lagrangian concordance, exact fillings, and rational point counts.
New dg-algebras link graph colorings to sheaves.
problem Linking graph colorings to sheaves for Legendrian surfaces.
method Generalized Casals-Murphy dg-algebra to non-commutative coefficients and computed Legendrian contact dg-algebra.
result Rank r representations of dg-algebras correspond to colorings of faces in Grassmannian.
Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second order differential equations on Euclidean space. One naturally wond…
We are enveloped by stories of visual interpretations in our everyday lives. The way we narrate a story often comprises of two stages, which are, forming a central mind map of entities and then weaving a story around them. A contributing factor to coherence is not just basing the story on these entities but also, refer…
Unified approach to learn interpretable concepts from data.
problem Building interpretable machine learning models and highly-performing foundation models.
method Relating causal representation learning and foundation models, defining concepts and proving their recoverability.
result Provable recovery of human-interpretable concepts from diverse data.
Sentiment analysis from news and social media predicts forex market movements.
problem Forecasting forex market movements using sentiment analysis.
method Lexicon-based analysis and Naive Bayes machine learning.
result Sentiment analysis is effective in predicting forex market movements.
A new polynomial invariant for links in a thickened torus exhibits volume conjecture behavior.
problem Defining and characterizing a new polynomial invariant for links in a thickened torus.
method Defining a new invariant , proving properties, and providing constructions.
result The invariant exhibits volume conjecture behavior, providing the first example of this in a virtual link.
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
The paper connects Legendrian links to cluster algebras via microlocal methods.
problem Understanding the relationship between Legendrian links and cluster algebras.
method Microlocal parallel transport of sheaf quantizations of Lagrangian fillings.
result Existence of quasi-cluster -structures and cluster Poisson structures.
New theory predicts deep neural networks can operate in an extended critical regime without fine-tuning.
problem Understanding the dynamics and computational principles of deep neural networks.
method Combining theories of heavy-tailed random matrices and non-equilibrium statistical physics.
result Deep neural networks can operate in an extended critical regime without fine-tuning parameters.
Improved Monte-Carlo models by constraining mutual information between latent and observable variables.
problem Training density models leads to latent variables being useless.
method Weave tighter Monte-Carlo bounds with mutual information constraints.
result Improved training of models with continuous and discrete latent variables.
Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. This paper takes the perspect…
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.
Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applicationsstat.OT
(The third edition corrects minor typos and adds 3 chapters synthesized from published papers plus an appendix on maximum entropy distributions.) The monograph investigates the misapplication of conventional statistical techniques to fat tailed distributions and looks for remedies, when possible. Switching from thin ta…
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.