Generalizes meander diagrams to virtual knots and introduces new invariants.
problem Classifying and comparing virtual knots.
method Generalization of meander diagrams to virtual knots, introduction of virtual k-arc crossing numbers. result Classes of meander and semimeander diagrams are universal for virtual knots.
Resolving Schwartz's quadratic meander number conjecture
problem Meander number of cyclic permutations
method Constructing families of cyclic permutations
result Meander number is bounded above and below quadratically in n
Counting meanders on surfaces of arbitrary genus, with precise asymptotics.
problem Counting and understanding meanders on surfaces of arbitrary genus.
method Square-tiled surfaces, moduli spaces of Abelian and quadratic differentials, Witten-Kontsevich 2-correlators.
result Asymptotic probability and polynomial growth of meanders with intersections.
A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in par…
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.
Paper solves the realizability of Gauss diagrams and constructs meanders.
problem Realizing Gauss diagrams as plane curves.
method Direct approach using conditions based on exits, entrances, and Jordan curve theorem.
result Conditions for realizability of Gauss diagrams and an algorithm to construct meanders.
Straight numbers generalize Meander and OGC numbers for all knots.
problem Defining invariants for all knots based on Meander and OGC numbers.
method Generalized Meander and OGC numbers to all knots and proved their well-definedness.
result Straight numbers and contained straight numbers are well-defined for all knots.
We introduced concept of meander knots, 2-component meander links and multi-component meander links and derived different families of meander knots and links from open meanders with at most 16 crossings. We also defined semi-meander knots (or knots with ordered Gauss code) and their product.
Study bounds growth rate of irreducible meanders, showing proportion vanishes.
problem Understanding growth rate of irreducible meanders.
method Provided upper and lower bounds for growth rate.
result Proportion of irreducible meanders among prime meanders approaches 0 as n increases.
Study on a new class of meanders with tangential intersections.
problem Enumerating and understanding meanders with transverse intersections.
method Developed a combinatorial framework, identified connections with other objects, and enumerated specific families.
result Completely enumerated several families of singular meanders.
New estimate of semimeander complexity for knots with more than 10 crossings.
problem Estimating the complexity of semimeander diagrams of knots.
method Proved a new upper bound on the number of crossings for semimeander diagrams of knots with more than 10 crossings.
result For knots with more than 10 crossings, semimeander diagrams have no more than 0.31⋅1.558cr(K) crossings. Paper proves C0-semi-rigidity of meandering-hyperbolic actions.
problem Stability of meandering-hyperbolic actions in C0 topology. method Proves semi-rigidity using local C0-semi-rigidity. result Every meandering-hyperbolic action is locally semi-rigid in C0 topology. New stability theorem for non-hyperbolic group actions.
problem Structural stability of non-hyperbolic group actions.
method Introducing 'meandering hyperbolicity' for group actions on geodesic metric spaces.
result Meandering-hyperbolic actions are structurally stable.
A meander of order n is a simple closed curve in the plane which intersects a horizontal line transversely at 2n points. (Meanders which differ by an isotopy of the line and plane are considered equivalent.) Let Gamma_n be the Cayley graph of the symmetric group S_n as generated by all (n choose 2) transpositions. Let …
Proves generalized meander conjectures for knots and spatial graphs.
problem Proving generalized meander conjectures for knots and spatial graphs.
method Study decomposition into simple arcs for diagrams of knots and spatial graphs.
result Proves generalized Jablan--Radović conjectures for knots and spatial graphs.
Minimal constructions of meanders and hyperelliptic pillowcase covers help in understanding ratio-optimizing pseudo-Anosovs.
problem Understanding ratio-optimizing pseudo-Anosovs in moduli spaces of quadratic differentials.
method Minimal constructions of meanders and hyperelliptic pillowcase covers.
result Existence of ratio-optimizing pseudo-Anosovs deep in the Johnson filtration.
We found a way to code meanders and show they are idempotent.
problem Understanding and coding meandric permutations.
method We established a bijection between meanders and Gauss diagrams, and used this to construct matrices that are idempotent.
result Meandric permutations are idempotent over the field GF(2).
Universal knot diagrams found in potholder curves.
problem Representing all knots and links efficiently.
method Comparing efficiency of curve types.
result Potholder curves represent all knots.
The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
problem Problems with previous descriptions of realizable Gauss diagrams.
method Experimental checking and formulation of new descriptions of realizable circle graphs.
result New descriptions of realizable circle graphs and an algorithm for checking realizability.
We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the conditional diffusion limit of the price process is the so-called Brownian meander.
This note re-addresses the Paris barrier options proposed by Yor and collaborators and their valuation using the Laplace transform approach. The notion of Paris barrier options, based on excursion theory and using the Brownian meander, is extended such that their valuation is now possible at any point during their life…
Paper presents a GAN model for realistic river image synthesis.
problem Generating high-quality river images for hydrological research.
method Used a Progressive Growing GAN (PGGAN) architecture to overcome training challenges.
result Demonstrated the effectiveness of GANs in generating high-resolution river images.
Investigates Q value evolution in Stable Baselines for DQL in simple vs complex environments.
problem DQL in Stable Baselines struggles with simple non-game environments.
method Comparison of TrafficLight and FrozenLake environments; Q value decomposition analysis.
result Q values meander far from optimal in complex relationships between states.
Study uses DMD to analyze oceanic features in Strait of Gibraltar.
problem Understanding complex oceanic features in Strait of Gibraltar.
method Dynamic Mode Decomposition (DMD) applied to 3D MIT general circulation model simulations.
result Unveiled new elements and dynamics of the Strait of Gibraltar, including a secondary gyre and wave propagation.
Study on knot diagrams showing bridge number can differ from crossing number.
problem Incompatibility between crossing number and bridge number for knot diagrams.
method Defined and compared various bridge number computations for knot diagrams, studied minimizing diagrams, and constructed families of minimal crossing diagrams.
result Found examples where bridge number differs from crossing number, and demonstrated this difference can grow infinitely.
Spatial embeddings of planar graphs can have higher unknotting numbers than crossing numbers.
problem Understanding the relationship between unknotting numbers and crossing numbers of spatial embeddings of planar graphs.
method Analyzing specific examples of planar graphs and their spatial embeddings to find counterexamples.
result There exist planar graphs and their spatial embeddings where the unknotting number is greater than half the crossing number.
The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Math…
The paper bounds the handle number of sutured manifolds using Morse-Novikov numbers and tunnel numbers.
problem Bounding the handle number of sutured manifolds.
method Developed bounds on the Morse-Novikov number of a link in terms of its tunnel number, and used these to bound the handle number of Heegaard splittings.
result The handle number function is bounded, constant on rays from the origin, and locally maximal.
New measure shows how links can be untangled as twists increase.
problem Understanding how links can be simplified through repeated twists.
method Introduced the stable unknotting number to analyze links in a twist family.
result The stable unknotting number depends only on the winding number of the link, not the wrapping number.
Wirtinger number equals virtual bridge number for virtual links.
problem Calculating the virtual bridge number of virtual links.
method Algorithmically computing the minimum number of generators of the link group.
result The Wirtinger number equals the virtual bridge number for virtual links.
The study provides bounds and necessary conditions for tunnel and cutting numbers of knots and handlebody-knots.
problem Determining bounds for tunnel and cutting numbers of knots and handlebody-knots.
method Using G-family of quandles colorings and constructing handlebody-knots.
result Lower bounds and necessary conditions for tunnel and cutting numbers of knots and handlebody-knots.
New number bounds knot complexity, including unknotting and crosscap numbers.
problem Bounding knot complexity and understanding knot types.
method Introducing an unknotting-type number to estimate crosscap number.
result Determines set of knots with crosscap number at most two.
New invariant refines Milnor's triple linking number, revealing more information for complex links.
problem Indeterminacy of Milnor's triple linking number in complex link configurations.
method Introduced a new invariant called the total triple linking number, refining Milnor's original.
result The total triple linking number is non-trivial for every (n≥6)-component link, providing more information than classical triple linking numbers. We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation…
Paper shows that for torus knots, the pinch number equals the unoriented band unknotting number.
problem Determining the minimum number of band surgeries to unknot torus knots.
method Used the torsion order of unoriented knot Floer homology.
result Pinch number and unoriented band unknotting number coincide for torus knots.
The paper studies tunnel and bridge numbers of composite genus 2 spatial graphs.
problem Understanding the tunnel and bridge numbers of composite genus 2 spatial graphs.
method Analyzes connected sum and trivalent vertex sum operations on genus 2 spatial graphs, proving bounds for tunnel and bridge numbers.
result Sharp bounds for the tunnel number of composite genus 2 spatial graphs, including lower bounds for bridge numbers.
Delta-unlinking number measures how to unlink algebraically split links.
problem Measuring unlinking complexity of algebraically split links.
method Defining delta-unlinking number as minimum delta-moves to unlink, proving bounds and calculating specific values.
result Precise delta-unlinking numbers for algebraically split prime links up to 9 crossings, and 4-genus values for most.
Study confirms a knot's crosscap number equals its splice-unknotting number for alternating knots.
problem Determining the crosscap number of alternating knots.
method Using a splice-unknotting number defined by Ito-Takimura, and computing through Gauss codes.
result Crosscap numbers of all prime alternating knots up to 13 crossings are computed.
Study on knot properties, showing relation between unknotting and crossing numbers.
problem Relations between unknotting and crossing numbers of spatial embeddings.
method Analyzes handcuff-graphs and theta curves, extends known results to handlebody-knots.
result Characterizes handlebody-knots satisfying the equality between unknotting and crossing numbers.
Odd crossing numbers and even rotation numbers for cycles in plane immersions.
problem Analyzing crossing and rotation numbers of cycles in plane immersions of graphs.
method Generic immersions and Legendrian embeddings of graphs, focusing on cycles of specific lengths.
result Sum of rotation numbers of all 5-cycles is even, and sum of crossing numbers is odd.
Lower bound found for Perron-Frobenius degrees of certain complex numbers.
problem Finding lower bounds for the Perron-Frobenius degree of complex numbers.
method Using Doug Lind's idea, proving results for both cubic and biPerron numbers.
result Arbitrary large Perron-Frobenius degrees for certain complex numbers.
This paper calculates stick numbers for rail arcs and knot classes.
problem Calculating the minimum number of sticks needed for rail arcs and knot classes.
method Rail isotopies, ambient isotopies, winding number invariant, and lattice stick number.
result Calculates stick numbers for rail arcs and knot classes with crossing number at most 9.
The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number were principally no more than linear in the number of classic…
Study computability of real numbers from group properties.
problem Computability of real numbers from group properties.
method Analyzing L2-Betti numbers and L2-torsion of groups. result Real numbers as L2-Betti numbers or L2-torsion are computable. We define the basket number, the flat plumbing number and the flat plumbing basket number of a link. Then we provide some upperbounds for these plumbing numbers by using Seifert's algorithm. We study the relation between these plumbing numbers and the genera of links.
The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the follow…
We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes to a slice knot; and the concordanc…
In this paper we investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinkin…