arXiv research

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.

A region crossing change at a region of a spatial-graph diagram is a transformation changing every crossing on the boundary of the region. In this paper, it is shown that every spatial graph consisting of theta-curves can be unknotted by region crossing changes.

Paper classifies link diagrams on nonorientable surfaces using region crossing changes.

Region crossing change is a local operation on link diagrams. The behavior of region crossing change on $S^2$ is well understood. In this paper, we study the behavior of (modified) region crossing change on higher genus surfaces.

Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.

Study examines how changing regions affects planar graphs.

In this paper, we prove that region crossing change on a link diagram is an unknotting operation if and only if the link is proper. A description of the behavior of region crossing change on link diagrams is given. Furthermore we also discuss the relation between region crossing change and the Arf invariant of proper l…

Region crossing change is a local transformation on a knot or link diagram. We show that a region crossing change on a knot diagram is an unknotting operation, and we define the region unknotting numbers for a knot diagram and a knot.

New game defined on origami patterns, linking number introduced.

New Boolean algebra method shows knot unknotting number is (c+1)/2.

Generalizes region select game to $k$-colored knot diagrams.

In a recent work of Ayaka Shimizu$^{[5]}$, she defined an operation named region crossing change on link diagrams, and showed that region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that region crossing change on a 2-component link diagram is an unknotting operation if and only…

Study on bit threads and their locking properties in holographic spacetimes.

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover o…

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing cha…

We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinat…

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…

In this paper we propose {\it a region choice problem} for a knot projection. This problem is an integral extension of Shimizu's 'region crossing change unknotting operation.' We show that there exists a solution of the region choice problem for all knot projections.

The study finds lower bounds for the warping degree of a knot projection.

The paper defines new polynomials for links and linkoids.

Proposes a method to solve deep neural networks' local minimum problem.

Simple regional perturbations maintain model transferability while reducing adversarial example distortion.

Region crossing change for a knot or a proper link is an unknotting operation. In this paper, we provide a sharp upper bound on the region unknotting number for a large class of torus knots and proper links. Also, we discuss conditions on torus links to be proper.

The paper identifies all link projections with isolate-region number one.

Paper speeds up visualization of uncertain data.

The DAO Report led to a significant shift of ICO activity to Europe.

Every knot has a plat projection, obtained by closing up a braid with bridges. The plat projection is determined by the number of strands and the number of rows of twist regions in the braid, and an integer number of crossings in each twist region. In recent work, we showed that under certain restrictions, including th…

The study refines algebraic domains with specific boundary conditions.

This study presents an extension of the Gaussian process regression model for multiple-input multiple-output forecasting. This approach allows modelling the cross-dependencies between a given set of input variables and generating a vectorial prediction. Making use of the existing correlations in international tourism d…

Problems on region choices for knot and link diagrams solved using Alexander numbering.

The Blum medial axis rigidity is studied in terms of cross ratios and differential geometry.

Proposes TCV for selecting models in regions of interest.

The study finds significant power-law cross correlations in Bitcoin's return-volatility dynamics.

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is call a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynom…

In this paper an approach based on expectation maximization (EM) clustering to find the climate regions and a support vector machine to build a predictive model for each of these regions is proposed. To minimize the biases in the estimations a ten cross fold validation is adopted both for obtaining clusters and buildin…

The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics …

Cross-domain collaborative filtering (CF) aims to alleviate data sparsity in single-domain CF by leveraging knowledge transferred from related domains. Many traditional methods focus on enriching compared neighborhood relations in CF directly to address the sparsity problem. In this paper, we propose superhighway const…

We present two families of knots which have straight number higher than crossing number. In the case of the second family, we have computed the straight number explicitly. We also give a general theorem about alternating knots that states adding an even number of crossings to a twist region will not change whether the …

We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions…

We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular we consider the case of component multi-tribrackets which have different tribracket operations at single-component crossings and multi-componen…

New formula for rotation number without needing a base point.

The expectation-maximization (EM) algorithm has been widely used in minimizing the negative log likelihood (also known as cross entropy) of mixture models. However, little is understood about the goodness of the fixed points it converges to. In this paper, we study the regions where one component is missing in two-comp…

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…

Analysis finds no evidence of banks managing deposit run risk prior to 2023 Regional Banking Crisis.

Core groups are link invariants defined by arc or region presentations.

We show that fMRI analysis using machine learning tools are sufficient to distinguish valence (i.e., positive or negative) of freely retrieved autobiographical memories in a cross-participant setting. Our methodology uses feature selection (ReliefF) in combination with boosting methods, both applied directly to data re…

Introduces CCR for constructing confidence regions from conformal predictions.

Adam achieves optimal convergence in deep ReLU networks via novel Kakeya bounds.