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.
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48 results for “crossing change”
Paper classifies link diagrams on nonorientable surfaces using region crossing changes.
problem Classifying link diagrams on nonorientable surfaces.
method Classification through region crossing changes.
result Classification of link diagrams on nonorientable surfaces.
Region crossing change is a local operation on link diagrams. The behavior of region crossing change on is well understood. In this paper, we study the behavior of (modified) region crossing change on higher genus surfaces.
The paper explores when specific knot operations simplify diagrams.
problem Understanding when arc crossing changes simplify knot diagrams.
method Examined two types of arc crossing changes on link diagrams and determined when they are unknotting operations.
result Any two crossing points in an alternating knot diagram are arc crossing change admissible.
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.
Alexander polynomial condition blocks crossing changes in some knots.
problem Cosmetic crossing changes in knot diagrams.
method Alexander polynomial obstruction for -space knots.
result Proved the conjecture for a five-parameter family of pretzel knots.
Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.
problem Unknottability of spatial graphs by region crossing changes.
method Region crossing changes to switch over/under relations within regions of spatial graph diagrams.
result Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.
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…
Study examines how changing regions affects planar graphs.
problem Effect of region crossing change on planar trivalent graphs.
method Investigation of region crossing changes on planar trivalent graphs.
result Effect of region crossing change on planar trivalent graphs.
A crossing in a knot is nugatory if changing the crossing does not change the knot type. Using an invariant of certain types of closed 3-braid diagrams, we show that if a closed 3-braid contains a nugatory crossing then its braid index is one or two. This proves a special case of a conjecture on nugatory crossings due …
We show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| \geq 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered ca…
Proves special alternating knots can't have cosmetic crossings.
problem Cosmetic crossing changes in special alternating knots.
method Combines Lidman-Moore and Scharlemann-Thompson results.
result Special alternating knots do not admit non-nugatory crossing changes.
Shows large unknotting number for simple knots.
problem Finding minimum crossing changes for unknotting.
method Positive-to-negative crossing changes without increasing genus.
result Genus non-increasing totally positive unknotting number can be large.
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 method detects changes by maximizing cross-entropy, outperforming existing techniques.
problem Detecting abrupt changes in data streams without labeled examples.
method Maximizes cross-entropy between segments to find change points, using dynamic programming.
result Outperforms three state-of-the-art approaches on challenging datasets.
The purpose of this article is to give a preliminary clarification on the relation between crossing number and crossing change. With a main focus on the span of X polynomial, we prove that, as our theorem claims, the crossing number of the link after crossing change can be estimated when certain conditions are met. At …
Extends knot invariant to filtered grid complexes.
problem Knot invariants and grid complexes.
method Combining Ozsváth-Szabó-Stipsicz crossing-change maps with Alishahi-Eftekhary l(K) invariant.
result Combinatorial formulation of knot invariant.
In this paper we define Crossing Change Alternating Knots (CCA knots) and their generalization: -CCA knots.
We analyze all monodromies of genus one fibered knots that possess clean or once-unclean arcs, and use this to determine all manifolds containing genus one fibered knots with generalized crossing changes resulting in another genus one fibered knot, and classify all such generalized crossing changes between two genus on…
In a recent work of Ayaka Shimizu, 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…
New game defined on origami patterns, linking number introduced.
problem Defining a game on origami patterns.
method Introduced Region Select on origami crease patterns.
result Defined a new unlinking number.
We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.
Colored Unlinkingmath.GT
In links with two components there are three different types of crossings: self-crossings in the first component, self crossings in the second component, and crossings between components. In this paper we examine the minimum number of crossing changes needed to unlink without changing the crossings between components. …
Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-…
We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes bet…
Graphs represent knot adjacency for n crossings.
problem Understanding adjacency relationships between knots.
method Defined a new graph to represent -adjacency.
result Proved several results about the new graph .
We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of forming certain satellites of winding number zero. We also define strongly cosmetic crossing changes and we discuss their behavior under the operation of inserting full twists in the strings of closed braids.
Cross-validation pitfalls in change-point regression are addressed with new approaches.
problem Cross-validation's prediction error-based criterion may lead to under- or over-estimation of change-points.
method Proposes two approaches: absolute error loss and modified holdout sets.
result Consistent estimation of the number of change-points under certain conditions.
We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and then unknot the resulting components.
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…
We use technology from sutured manifold theory and the theory of Heegaard splittings to relate genus reducing crossing changes on knots in S^3 to twists on surfaces arising in circular Heegaard splittings for knot complements. In a separate paper, currently in preparation, we prove that these circular Heegaard splittin…
New Boolean algebra method shows knot unknotting number is (c+1)/2.
problem Finding the minimum number of region crossing changes to unknot a knot.
method Boolean algebra applied to region crossing changes.
result Region unknotting number is (c+1)/2 for any knot with crossing number c.
A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the we…
We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can therefore also determine whether the unlinking number of L is 1.
A cobordism realizing crossing change on tangle homology and a categorified Vassiliev skein relationmath.GT
The paper introduces a cobordism for Khovanov homology crossing change and categorifies Vassiliev skein relations.
problem Khovanov homology and crossing changes in tangle diagrams.
method Introducing a sum of cobordisms that yields a morphism on Khovanov homology complexes for crossing change.
result The introduced cobordism is invariant under double point moves and categorifies Vassiliev skein relations.
Agent finds unknotting sequences for complex knots.
problem Finding minimal crossing changes to unknot complex knots.
method Reinforcement learning agent for knot diagrams.
result Upper bounds on unknotting numbers for thousands of knots.
Proves links can be simplified to trivial form in few changes, limiting Milnor's invariants.
problem Limits of Milnor's invariants in understanding unlinking number.
method Sequence of links, crossing changes, and geometric filtration.
result Crossing changes to -trivial link grows quadratically with number of components.
The study eliminates infinite families of knots with nontrivial Alexander polynomials and improves unknotting number data.
problem Identifying knots with nontrivial Alexander polynomials and improving knot classification.
method Elimination of infinite families of knots and use of determinants to improve unknotting number data.
result Elimination of infinite families of knots with nontrivial Alexander polynomials and improvement of unknotting number data.
In this paper, we discuss the crossing change operation along exchangeable double curves of a surface-knot diagram. We show that under certain condition, a finite sequence of Roseman moves preserves the property of those exchangeable double curves. As an application for this result, we also define a numerical invariant…
New findings on knot operations challenge a long-standing conjecture.
problem Understanding equivariant unknotting numbers of strongly invertible knots.
method Study of symmetric crossing change operations for strongly invertible knots.
result The equivariant unknotting number is not additive under connected sum.
The study examines cross-border lending behavior from G7 countries, showing changes in driving factors after the 2008 financial crisis.
problem Understanding the factors affecting cross-border lending behavior among G7 countries.
method Employed a gravity model to analyze bilateral and global factors influencing cross-border lending.
result Driving factors for cross-border lending have changed since the 2008 financial crisis, with continent variable becoming more significant.
A graph G is called "minimalizable" if a diagram with minimal crossing number can be obtained from an arbitrary diagram of G by crossing changes. If, furthermore, the minimal diagram is unique up to crossing changes then G is called "strongly minimalizable". In this article, it is explained how minimalizability of a gr…
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.
New polynomial invariants defined for long virtual knots.
problem Defining and studying polynomial invariants for long virtual knots.
method Intersection numbers of cycles on a closed surface, considering crossing order.
result Intersection polynomials are finite-type invariants of degree two under crossing changes, but not under virtualizations.
Unknotting genus one knotsmath.GT
For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are precisely two unknotting crossing changes. The proof uses sutured manifold theory and …
The slicing number of a knotmath.GT
An open question asks if every knot of 4-genus g_s can be changed into a slice knot by g_s crossing changes. A counterexample is given.
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…
Study sharpens unlinking number bounds for special alternating links.
problem Determining the exact unlinking number for special alternating links.
method Analyzes links in the 3-sphere, focusing on special alternating links and their crossing changes.
result Sharp lower bounds for unlinking number realized by crossing changes in alternating diagrams.