Classifies changes between genus one fibered knots.
problem Understanding changes between genus one fibered knots.
method Analyzed all monodromies and determined manifolds.
result Classified all generalized crossing changes between genus one fibered knots.
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…
Alexander polynomial condition blocks crossing changes in some knots.
problem Cosmetic crossing changes in knot diagrams.
method Alexander polynomial obstruction for L-space knots. result Proved the conjecture for a five-parameter family of pretzel knots.
The paper studies unlinking links with minimal crossing changes.
problem Unlinking links with minimal changes when crossings between components are fixed.
method Examined minimum number of crossing changes for links with linking number zero and unknotted components.
result Data and general results about asymmetry of unlinking between components for links with up to ten crossings.
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 how region crossing change behaves on surfaces of different genus.
problem Understanding region crossing change on surfaces of varying genus.
method Analyze (modified) region crossing change on higher genus surfaces.
result Behavior of region crossing change differs on higher genus surfaces.
In this paper we define Crossing Change Alternating Knots (CCA knots) and their generalization: k-CCA knots.
Spatial graphs can be unknotted with region crossing changes.
problem Unknotted spatial graphs composed of theta-curves.
method Region crossing changes on regions of theta-curves.
result Spatial graphs of theta-curves can be unknotted.
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…
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.
Graphs represent knot adjacency for n crossings.
problem Understanding adjacency relationships between knots.
method Defined a new graph Γn to represent n-adjacency. result Proved several results about the new graph Γn. 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.
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 …
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.
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.
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.
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…
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…
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.
The paper explores Gordian complexes of knots and virtual knots using region crossing changes and arc shift moves.
problem Defining and analyzing Gordian complexes of knots and virtual knots using specific local moves.
method Region crossing change and arc shift move to construct Gordian complexes.
result Existence of arbitrarily high dimensional simplices in both Gordian complexes.
Generalizes region select game to k-colored knot diagrams.
problem Play game on knot diagrams with multiple colors.
method Generalize region select game to k-colored knot diagrams. result Generalization of the region select game to k-colored knot diagrams. Study unlinking numbers of 10-crossing links using link invariants.
problem Investigate unlinking numbers of 10-crossing links.
method Use various link invariants and explore their behavior with crossing changes.
result Find the unlinking numbers of all but 2 of the 287 prime, non-split links with crossing number 10.
Algorithm determines unlinking number 1 for certain links.
problem Determining unlinking number 1 for links.
method Algorithm to find crossing changes turning links into split links.
result Algorithm determines unlinking number 1 for certain links.
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 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.
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.
Study of knots in homology spheres and their concordance properties.
problem When can knots in homology spheres be reduced to slice knots?
method Analyzes concordance and crossing changes in homology spheres.
result Knots in homology spheres are nullhomotopic in a smooth homology ball if and only if they are concordant to a smoothly slice knot.
Defines new invariant for virtual knots.
problem Determining unknottability of virtual knots.
method Parity virtual Alexander polynomial.
result Many virtual knots cannot be unknotted by odd crossing changes.
Study unknotting numbers of prime θ-curves up to 7 crossings.
problem Determine unknotting numbers for prime θ-curves.
method Subadditivity of unknotting numbers, non-overlapping set analysis, crossing changes, new methods for obstructing unknotting number 1.
result Exact unknotting numbers for all prime θ-curves up to 7 crossings.
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 new index measures how many changes are needed to turn virtual links into simple ones.
problem Determining the minimum number of changes needed to simplify virtual link diagrams.
method Defined the unknotting index using virtual crossings and classical changes, provided lower and upper bounds.
result Found bounds for the unknotting index of virtual links, including those from pretzel links.
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.
New families of knots with more straight segments than crossings.
problem Understanding knots with more straight segments than crossings.
method Presented two families of knots, computed straight number for one, and provided a theorem about alternating knots.
result Explicit computation of straight number for one family of knots and a general theorem about alternating knots.
Study calculates site-specific Gordian distances between graph embeddings.
problem Determining the minimal number of crossing changes between graph embeddings.
method Covering space theory for proofs.
result Site-specific Gordian distances between Milnor links and trivial links are determined.
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.
New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.
problem Stable generalized complex structures in higher dimensions with self-crossing singularities.
method Extending stable generalized complex structures to include anticanonical sections with normal self-crossings.
result Construction of large families of stable generalized complex manifolds in four dimensions.
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 Ck-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…