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…
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48 results for “knot unknotting”
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 invariant measures how many twists are needed to unknot welded knots.
problem Measuring complexity of welded knots.
method Local twist move, Alexander quandle coloring, Gordian distance.
result Established lower bound on twist number and related it to other knot invariants.
The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this note we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if …
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.
Study on bounds of knot untangling for specific types of knots.
problem Determining upper limits for knot untangling.
method Defined warping degree, examined diagrams combinatorially.
result Upper bounds for unknotting and region unknotting numbers.
The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings o…
Study determines -unknotting numbers for two-bridge knots.
problem Determining the minimum number of -moves to simplify two-bridge knots.
method Examined two-bridge knots of specific types and calculated their -unknotting numbers.
result Identified two-bridge knots with -unknotting number equal to one.
Lower bounds on unknotting number for cabled knots.
problem Difficulty in computing unknotting number and understanding its behavior under cabling.
method Combining knot Floer homology bounds with computations of cable knot Floer homology.
result Established a lower bound on the unknotting number of cable knots in terms of winding number.
We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alte…
Two algorithms use normal surfaces to detect unknots and prove knots.
problem Detecting and proving the unknot and knottedness of links.
method Normal surface theory algorithms and split-link algorithm.
result Figure-eight knot is proven to be knotted.
New method for simplifying knots with specific properties.
problem Understanding knots with a specific unknotting number.
method Derive and apply the Montesinos trick for proper rational tangle replacement.
result Prove that knots with proper rational unknotting number one are prime and classify certain types.
Knot 11n102 requires 2 changes to untangle.
problem Determining the minimum number of changes needed to untangle a knot.
method Analyzing the knot 11n102 to find its unknotting number.
result The unknotting number of 11n102 is 2.
New diagonal move simplifies knots and links efficiently.
problem Efficiently unknotting knots and links.
method Introduces diagonal move, proving its effectiveness for classical and welded knots.
result Diagonal move reduces any knot or link to the unknot or unlink with fewer operations.
The paper calculates the Δ-unknotting number for positive pretzel knots.
problem Determining the minimum number of Δ-moves to simplify pretzel knots.
method Analyzing the Conway polynomial and computing Δ-unknotting numbers.
result The Δ-unknotting number for positive pretzel knots is maximized by a specific knot type.
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.
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.
Grid homology shows knot unknotting lower bound.
problem Knot unknotting number determination
method Grid homology analysis
result Torsion homology classes order bounds unknotting number
We prove new results about unknotting fibered positive knots and braids.
problem Proving the unknotting number equals genus for fibered positive knots and braids.
method Analyzing positive braid diagrams and fibered positive knots, proving new constraints and conjectures.
result We found fibered positive knots that cannot be unknotted optimally, contradicting Stoimenow's conjecture.
We determine a wide class of knots, which includes unknotting number one knots, within which Khovanov homology detects the unknot. A corollary is that the Khovanov homology of many satellite knots, including the Whitehead double, detects the unknot.
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.
Single twist can unknot certain knots, study shows.
problem Can a knot be unknotted with a single twist?
method Classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.
result Obstructions to unknotting with a single twist exist.
RL pipeline simplifies knot diagrams, including very hard unknots.
problem Simplifying complex knot diagrams, especially very hard unknots.
method Reinforcement learning for move proposals and heuristic navigation of Reidemeister moves.
result Trained agent simplifies diagrams, including a link to a three-step unknotting process.
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…
A knot in the 3-sphere is said to have zero negative unknotting number if it can be transformed into the unknot by performing only positive crossing changes. In this paper, we provide an obstruction for a knot to having zero negative unknotting number, and discuss its application to two classes of knots.
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.
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.
In this paper, the authors give an unknotting sequence for torus knots and also provide unknotting numbers of and some other knots from the knot table of Hoste-Thistlethwite.
In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In particular, we show that for any non-negative integer there exists a virtual knot w…
Smoothly slice a knot with specific properties.
problem Existence of certain types of knots.
method Proving the existence of a specific knot with given properties.
result Existence of a smoothly doubly slice, amphicheiral knot with Alexander polynomial 1 and unknotting number 5.
Null-homologous unknottingsmath.GT
Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.
Knots 4_1 and 5_1 can be paired to show unknotting number is not additive.
problem Exploring the additivity of the unknotting number for specific knots.
method Pairing knots and analyzing their connected sums.
result Unknotting number is not additive for knots 4_1 and 5_1.
The unknotting number is the classical invariant of a knot. However, its determination is difficult in general. To obtain the unknotting number from definition one has to investigate all possible diagrams of the knot. We tried to show the unknotting number can be obtained from any one diagram of the knot. To do this we…
The paper finds virtual knots with specific unknotting indices.
problem Finding virtual knots with unknotting indices (n,m).
method Providing infinite families of virtual knots and links with specified unknotting indices.
result Existence of infinitely many distinct virtual knot diagrams with unknotting index (n,m).
We use the rational Witt class of a knot in the 3-sphere as a tool for addressing questions about its unknotting number. We apply these tools to several low crossing knots (151 knots with 11 crossing and 100 knots with 12 crossings) and to the family of n-stranded pretzel knots for various values of n>2. In many cases …
We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number…
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.
Vertex distortion detects if a knot is unknot.
problem Determining if a knot is the unknot.
method Using Denne-Sullivan's bound on Gromov distortion, the vertex distortion of nontrivial lattice knots is bounded. Then, it is shown that trivial vertex distortion implies the unknot.
result The conjecture that trivial vertex distortion implies the unknot is proven.
The H(n)-move simplifies virtual and welded knots and links.
problem Tackling the unknotting of virtual and welded links.
method Extending the H(n)-move to virtual and welded links and showing their equivalence to Reidemeister moves.
result Virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and H(n)-moves.
A bound on knot unknotting using equivariant signature.
problem Equivariant unknotting of knots.
method Analysis of strongly invertible knots and application of equivariant unknotting moves.
result The equivariant signature provides a lower bound for the equivariant unknotting number.
For a knot K in S^3, let T(K) be the characteristic toric sub-orbifold of the orbifold (S^3,K) as defined by Bonahon and Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EM-knot (of Eudave-Munoz) or (S^3,K) contains a…
The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.
problem Understanding the topology and symmetry of cylindrical handlebody-knots of genus two.
method Analysis of Thurston's hyperbolization theorem and investigation of unknotting annuli.
result The symmetry group is trivial if the unknotting annulus is unique and of type .
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 …
Characterizes unknotted curves on Seifert surfaces of twist knots.
problem Identifying unknotted curves on Seifert surfaces of twist knots.
method Analyzing homologically essential simple closed curves on Seifert surfaces of genus one knots.
result Characterizes unknotted curves on Seifert surfaces of twist knots, including infinitely many for the figure eight knot and one for Whitehead doubles.
A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is n-adjacent to the unknot, for some n>0, if and only if it is n-adjacent to the un…
Unknotting Unknotsmath.GT
A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of unknotting a knot diagram when we know that it is unknotted. This problem is surp…
New knot invariant λ bounds rational unknotting.
problem Bounding rational unknotting number.
method Extracted invariant λ from Khovanov homology.
result Invariant λ is lower bound for proper rational unknotting number.
The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting nu…