In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtu…
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48 results for “region arc shift move”
Study on unknotting twisted knots using arc shift and region arc shift moves.
problem Unknotting twisted knots and finding bounds for region arc shift number.
method Introduced arc shift move and region arc shift move for twisted knots.
result Found families of twisted knots with specific arc shift and region arc shift numbers.
Gordian complexes of knots and virtual knots given by region crossing changes and arc shift movesmath.GT
Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in . Later Horiuchi and Ohyama defined Gordian complex of virtual knots using -move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing cha…
The paper classifies virtual links using the arc shift operation.
problem Classifying \( n \)-component virtual links up to arc shift equivalence.
method Established the arc shift operation as an unknotting tool for \( n \)-homogeneous proper virtual links, explored its connection to the odd writhe, and identified sequences with specific arc shift bounds.
result Identified sequences of virtual link diagrams \( L_n \) with an upper bound of arc shift number equal to \( n \).
Shifts are not type-preserving on surface graphs.
problem Understanding the type-preserving property of shift maps on surface graphs.
method Analyzing Dehn twists and shift maps on arc, curve, and relative arc graphs of surfaces.
result Shift maps are not type-preserving on surfaces with isolated punctures.
The paper describes topological properties of arcs and crossings in knot theory.
problem Understanding the topological nature of arcs and crossings in knot theory.
method Topological description of arcs and crossings as isotopy classes of probes, homotopy classes of diagram elements.
result Sets of arcs and crossings are fundamental for algebraic objects like quandles, partial ternary quasigroups, biquandloids, and crossoids.
Improves arc separation result for homogeneous spaces.
problem Separating regions in homogeneous spaces by arcs.
method Using homogeneity instead of strong local homogeneity, and considering arcs with one interior point.
result Regions in homogeneous spaces of dimension ≥ 2 are not separated by arcs.
We work with a generalization of knot theory, in which one diagram is reachable from another via a finite sequence of moves if a fixed condition, regarding the existence of certain morphisms in an associated category, is satisfied for every move of the sequence. This conditional setting leads to a possibility of irreve…
Core groupsmath.GT
Core groups are link invariants defined by arc or region presentations.
problem Defining link invariants using different presentations of arcs and regions.
method Introducing core groups as link invariants defined by presentations involving arcs or regions, and extending these to virtual link diagrams.
result Properties of core groups and their extensions to virtual link diagrams are discussed.
We prove that any arc-presentation of the unknot admits a monotonic simplification by elementary moves; this yields a simple algorithm for recognizing the unknot. We obtain similar results for split links and composite links.
The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.
problem Counting arcs on hyperbolic surfaces with boundaries and cusps.
method Asymptotic analysis of pure mapping class group orbits and arc lengths.
result The number of arcs of bounded length is asymptotically proportional to .
We prove the existence of complete minimal surfaces of genus g>1 which minimize the total curvature for their genus. Our method is first to identify this (Weierstrass high dimensional period) problem with the problem of finding a particular type of polygonal arc in the complex domain: the arc alternates between horizon…
A knot K in 1-bridge position with respect to a genus-g Heegaard surface in a 3-manifold can be moved by isotopy through knots in 1-bridge position until it lies in a union of n parallel genus-g surfaces tubed together by n-1 straight tubes, with K intersecting each tube in two arcs connecting the ends. We prove that t…
A -move is a local operation for links consisting in replacing two parallel arcs by four half twists. At the present time, it is not known if this induces an unkotting operation for knots. Studying the Dabkowski-Sahi invariant, we prove that any invariant of knots based on the fundamental group …
We study the connections between subsurface projections in curve and arc complexes in fibered 3-manifolds and Agol's veering triangulation. The main theme is that large-distance subsurfaces in fibers are associated to large simplicial regions in the veering triangulation, and this correspondence holds uniformly for all…
New infinite-type loxodromic elements found in surface mapping classes.
problem Identifying infinite-type loxodromic elements in mapping classes of surfaces.
method Constructing infinite families of mapping classes acting loxodromically on the relative arc graph.
result Explicit construction and characterization of infinite-type loxodromic elements.
How and why stock prices move is a centuries-old question still not answered conclusively. More recently, attention shifted to higher frequencies, where trades are processed piecewise across different timescales. Here we reveal that price impact has a universal non-linear shape for trades aggregated on any intra-day sc…
The paper defines normal forms for rational 3-tangles and shows a sequence of moves to transform one form to another.
problem Understanding and manipulating rational 3-tangles.
method Definition of normal forms and sequence of normal jump moves.
result There is a sequence of normal jump moves leading to equivalent normal forms of rational 3-tangles.
Paper simplifies proof of slide-equivalence in crown diagrams.
problem Proving slide-equivalence of crown diagrams.
method Diagrammatic approach to generalized shift moves.
result Slide-equivalence of vanishing cycles in crown diagrams.
We study b-arc foliation change and exchange move of open book foliations which generalize the corresponding operations in braid foliation theory. We also define a bypass move as an analogue of Honda's bypass attachment operation. As applications, we study how open book foliations change under a stabilization of the op…
A Morse 2-function is a generic smooth map from a manifold M of arbitrary finite dimension to a surface B. Its critical set maps to an immersed collection of cusped arcs in B. The aim of this paper is to explain exactly when it is possible to move these arcs around in B by a homotopy and to give a library of examples w…
Theory of link projections to 3-manifold spines, proving combinatorial moves for isotopic links.
problem Link equivalence on 3-manifold spines.
method Theory of link projections, combinatorial moves, Reidemeister Theorem.
result Crossingless projections and combinatorial moves for isotopic links.
Refining the notion of an ideal triangulation of a compact three-manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,a), where M is a three-manifold and a is a collection of properly embedded arcs. We also show that certain well-understood combinatorial moves are sufficient to relate …
We introduce a local move on a link diagram named a region freeze crossing change which is close to a region crossing change, but not the same. We study similarity and difference between region crossing change and region freeze crossing change.
Study uses open data to improve traffic emissions estimation.
problem Estimating accurate link level traffic emissions.
method Data-driven framework integrating MOVES, GPS, OSM, and satellite imagery.
result Neural network reduces RMSE by over 50% for key pollutants.
New invariant shows fifth move is unique and extends biquandle theory for surface-links.
problem Reproduce Yoshikawa's fifth move using other moves and isotopies.
method Develops a semi-invariant and biquandle-based invariant for surface-links.
result Yoshikawa's fifth move is unique and cannot be reproduced by other moves.
Method uses aggregate crop statistics to improve satellite-based crop type mapping.
problem Limited field-level crop labels for training satellite-based maps.
method Corrects classifier by accounting for shifts in crop type composition and feature means.
result Substantial improvements in overall classification accuracy, reducing misclassifications by 21.9% on average.
This paper examines how adversarial perturbations affect model performance and equilibrium learning.
problem Adversarial perturbations and covariate shifts impact model performance and equilibrium learning.
method Characterizes the extrapolation region in regression and classification, analyzes dynamics of adversarial learning games.
result Establishes two directional convergence results: a blessing in regression and a curse in classification.
This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are found among the integers modulo n. The choice of n depends upon the knot. Each typ…
The paper studies strict equivalence in multi-virtual linkoids with new invariants.
problem Understanding strict equivalence in multi-virtual linkoids.
method Utilizing multi-virtual knot theory, defining strict virtual linkoids, and studying invariants.
result New invariants for strict virtual linkoids are defined.
Circular transformation of the European steel industry renders scrap metal a strategic resourceq-fin.TR
European steel industry shifts to electric arc furnaces, reducing scrap imports and increasing competition.
problem Reducing CO2 emissions in the European steel industry through electric arc furnaces.
method Combining trade data with business intelligence to model the impact of EAF capacity on scrap trade.
result Scrap imports decrease as EAF capacity increases, highlighting the need for a new business ecosystem.
A lens cluster minimizes perimeter in the plane with given area constraints.
problem Minimizing perimeter in the plane with given area constraints.
method Analyzing lens clusters consisting of circular arcs with specific geometric properties.
result Lens clusters are local minimizers of the total perimeter functional.
K-color region select gamemath.GT
Generalizes region select game to -colored knot diagrams.
problem Play game on knot diagrams with multiple colors.
method Generalize region select game to -colored knot diagrams.
result Generalization of the region select game to -colored knot diagrams.
The paper introduces various canonical parameterizations for 2D-curved shapes.
problem Comparing unparameterized simple curves in the plane.
method Proposes diverse canonical parameterizations, including arc-length and curvature-based.
result Natural parameterizations correspond to physical movements and are geometric invariants.
A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in the Minkowski space, which can naturally produce the conformal invariants and the no…
In this paper, we study how the mean shift algorithm can be used to denoise a dataset. We introduce a new framework to analyze the mean shift algorithm as a denoising approach by viewing the algorithm as an operator on a distribution function. We investigate how the mean shift algorithm changes the distribution and sho…
In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter…
We provide initial seedings to the Quick Shift clustering algorithm, which approximate the locally high-density regions of the data. Such seedings act as more stable and expressive cluster-cores than the singleton modes found by Quick Shift. We establish statistical consistency guarantees for this modification. We then…
In this paper, we consider domino tilings of regions of the form , where is a simply connected planar region and . It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjace…
New method tackles MNAR missingness in domain adaptation.
problem Handling missingness in both source and target data.
method Reduces MNAR missingness to imputation problem, leveraging recent MNAR imputation methods.
result Developed a novel domain adaptation procedure for MNAR missingness shift.
In this thesis, we consider domino tilings of three-dimensional regions, especially those of the form . In particular, we investigate the connected components of the space of tilings of such regions by flips, the local move performed by removing two adjacent dominoes and placing them back in t…
We study the problem of finding strain-minimising stream surfaces in a divergence-free vector field. These surfaces are generated by motions of seed curves that propagate through the field in a strain minimising manner, i.e., they move without stretching or shrinking, preserving the length of their arbitrary arc. In ge…
RNA foldings and Stuck Knotsmath.GT
Study uses knot theory to model RNA foldings, emphasizing both entanglement and intrachain interactions.
problem Modeling RNA foldings considering both entanglement and intrachain interactions.
method Combines knot theory with embedded rigid vertex graphs to emphasize both entanglement and intrachain interactions of RNA foldings.
result Defines and computes a coloring counting invariant for stuck links, providing explicit computations for arc diagrams of RNA foldings.
We introduce a topological combinatorial game called the Region Smoothing Swap Game. The game is played on a game board derived from the connected shadow of a link diagram on a (possibly non-orientable) surface by smoothing at crossings. Moves in the game are performed on regions of the diagram and can switch the direc…
A bypass in the middlemath.GT
New proofs of h-principles in contact 3-manifolds.
problem Characterizing contractible bypasses in contact 3-manifolds.
method Topological characterization and disjoint bypass construction.
result New proofs of h-principles in overtwisted contact 3-manifolds.
The paper establishes a relation between knotoid crossing number and height.
problem Establishing a relation between knotoid crossing number and height.
method Using Reidemeister moves and isotopies, the paper proves a relation between the crossing number and height of a knotoid.
result The crossing number of a knotoid is at least twice its height.
This paper develops methods for obtaining distribution-free prediction regions for invariant representations.
problem Distributional shifts in machine learning models.
method Invariant risk minimization and weighted conformity scores.
result Proves the effectiveness of adaptive conformal intervals for uncertainty estimation.
The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. Many long-term correlated time series in real systems contain various trends. We investigate the effects of polynomial trends on the scaling behaviors and the performa…