Upper bounds and minimum ribbonlengths for folded ribbon knots are derived.
problem Understanding the ribbonlength of folded ribbon knots.
method Analysis of Kauffman's model of folded ribbon knots, upper bounds calculation, and minimization proofs.
result Minimum ribbonlength for a 3-stick unknot is 3√3, achieved by an equilateral triangle.
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
Study on hard Legendrian unknots using normal rulings.
problem Understanding the complexity of Legendrian unknots in knot theory.
method Using normal rulings to obstruct and construct hard unknot diagrams.
result Construction of infinitely many smoothly hard max-tb unknot diagrams with bounds on minimum possible writhe.
Study shows unknotting number of certain virtual torus knots equals standard torus knot's unknotting number.
problem Determining the unknotting number of virtual torus knots.
method Analyzing virtual knots derived from standard torus knots and counting crossing changes.
result Virtual unknotting number of certain virtual torus knots equals the unknotting number of the corresponding standard torus knot.
New SCI index bounds unknotting framed unknots.
problem Bounding unknotting framed unknots.
method Introducing Self-Crossing Index (SCI) and using it to provide bounds.
result SCI provides bounds for unknotting framed unknots.
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.
The paper explores different types of knot unknotting numbers using various local moves.
problem Investigating the unknotting numbers of knots using different local moves.
method Examined ribbon-move and pass-move on 2-knots and 1-knots, and high-dimensional-pass-move on high-dimensional knots.
result Found examples and bounds for various unknotting numbers associated with different local moves.
New knot invariant detects unknots.
problem Detecting unknots in knot theory.
method Introduced Alexander-Beck module as a refined Alexander module.
result Alexander-Beck module detects the unknot.
Proves Jones unknot conjecture for knots up to 23 crossings.
problem Verifying the Jones polynomial distinguishes the unknot from nontrivial knots.
method Analyzing Jones polynomials up to 23 crossings.
result Jones polynomial successfully distinguishes the unknot from nontrivial knots up to 23 crossings.
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.
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.
Paper introduces unknotting index for virtual knots.
problem Computing and understanding virtual knots.
method Using writhe invariants and relationships with virtual knot module.
result Existence of virtual knots with specific unknotting indices.
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.
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…
Grid homology shows knot unknotting lower bound.
problem Knot unknotting number determination
method Grid homology analysis
result Torsion homology classes order bounds unknotting number
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 41#910 link to a three-step unknotting process. 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.
This paper concerns the H(2)-unknotting numbers of links related to 2-bridge links. It consists of three parts. In the first part, we consider a necessary and sufficient condition for a 2-bridge link to have H(2)-unknotting number one. The second part concerns an explicit form of composite links with H(2)-unknotting nu…
Contractibility of unknot's weak reducing disks shown.
problem Contractibility of unknot's weak reducing disks in 3-bridge position.
method Analysis of weak reducing disks in 3-bridge position.
result Complex of weak reducing disks for the unknot in 3-bridge position is contractible.
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.
Three hard diagrams of the unknot require extra crossings to simplify.
problem Finding diagrams of the unknot that require many crossings to simplify.
method Applying previously proposed methods to construct diagrams and using computational resources to prove their hardness.
result Three hard diagrams of the unknot require at least three extra crossings.
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 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 …
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.
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.
Knots can be unknotted with specific twists, proving bounds on the number of twists needed.
problem The minimum number of twists needed to unknock a knot of a given genus.
method Proved using null-homologous twists, showing bounds on the number of twists required.
result There exist knots of genus g that cannot be unknotted with fewer than 2g null-homologous twists.
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.
Knot Floer homology provides bounds on knot unknotting numbers.
problem Bounding the unknotting number of knots.
method Using knot Floer homology to construct invariants l^-(K), l^+(K), and l(K).
result The invariant l(K) only vanishes for the unknot, and gives lower bounds on the unknotting number.
Paper finds knots that can't be simplified with only positive changes.
problem Identifying knots that can't be simplified with only positive crossing changes.
method Provides an obstruction for knots to have zero negative unknotting number.
result Knots with zero negative unknotting number can be identified.
Study shows nontrivial links as cross-sections of unknotted holomorphic disks.
problem Resolving parts of Kirby's list problem about nontrivial links.
method Using unknotted holomorphic disks in the four-ball to produce cross-sections.
result Many nontrivial links arise as cross-sections of unknotted holomorphic disks.
Study determines Δ-unknotting numbers for two-bridge knots.
problem Determining the minimum number of Δ-moves to simplify two-bridge knots. method Examined two-bridge knots of specific types and calculated their Δ-unknotting numbers. result Identified two-bridge knots with Δ-unknotting number equal to one. 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.
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.
Spatial embeddings of planar graphs can have higher unknotting numbers than crossing numbers.
problem Understanding the relationship between unknotting numbers and crossing numbers of spatial embeddings of planar graphs.
method Analyzing specific examples of planar graphs and their spatial embeddings to find counterexamples.
result There exist planar graphs and their spatial embeddings where the unknotting number is greater than half the crossing number.
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.
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.
This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is…
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.
New unknots with geometric constraints exist, proving a long-standing conjecture.
problem Existence of distinct isotopy classes of physical unknots with geometric constraints.
method Parametrised thickness and geometric thresholds to fragment isotopy classes.
result Existence of gordian unknots with prescribed geometric constraints.
New presentations and algorithms for links and virtual links simplify knot and unknot transformations.
problem Simplifying and transforming links and virtual links into simpler forms.
method Introducing new presentations, constructing algebraic systems, and proposing Reduction Crossing Algorithms.
result Known unknots and an infinite family of knots can be unknotted efficiently.
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).
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…
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.
We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsvath and Szabo's obstruction to unknotting number one. We determine the unknotting numbers of 9_10, 9_13, 9_35, 9_38, 10_53, 10_101 and 10_120; this completes the table of unknotting numbers for prime knots with crossi…
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 links found that aren't concordant to any with a single unknotted component.
problem Finding links that can't be concordant to any with a single unknotted component.
method Used Alexander modules.
result Constructed links of multiple components that are not concordant to any with a single unknotted component.
New measure shows how links can be untangled as twists increase.
problem Understanding how links can be simplified through repeated twists.
method Introduced the stable unknotting number to analyze links in a twist family.
result The stable unknotting number depends only on the winding number of the link, not the wrapping 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…