Unified method proves integrality of LMOV invariants for framed unknot.
problem Proving the integrality of LMOV invariants for framed unknots.
method Unified method to prove integrality of explicit formulae for LMOV invariants.
result Proves integrality of certain LMOV invariants for framed unknots in higher genera.
The paper calculates minimum crossing numbers for essential tangles.
problem Finding the minimum number of crossings in essential tangles.
method Computational analysis of tangles with string components.
result Characterization of tangles with minimum crossing numbers.
The paper examines when 2-string tangles can be embedded into specific link types.
problem When 2-string tangles can be embedded into the unknot, unlink, or split links.
method Geometric characterizations, tangle sums, and colorings.
result Prime 2-string tangles with up to seven crossings are classified for embedding into specific link types.
New method uses vortex strings to untangle knots.
problem Untangling complex knots.
method Reaction-diffusion dynamics of vortex strings in a nonlinear PDE.
result Evolution preserves knot topology and untangles unknots.
Study LMOV invariants for a framed unknot in toric Calabi-Yau 3-folds.
problem Computing LMOV invariants for a specific topological string setup.
method Explicit computation and reformulation of LMOV invariants for genus 0 and higher.
result Proved integrality of LMOV invariants and reformulated higher genus invariants.
New knot invariant derived from string topology.
problem Detecting the unknot using knot contact homology.
method String topology and Legendrian contact homology.
result Knot contact homology detects the unknot.
After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point parti…
We continue the study of the genus of knot diagrams, deriving a new description of generators using Hirasawa's algorithm. This description leads to good estimates on the maximal number of crossings of generators and allows us to complete their classification for knots of genus 4. As applications of the genus 4 classifi…
We prove all knots can be transformed into a trefoil using special diagrams.
problem Transforming any knot into a trefoil using magic tricks.
method Introducing knotholder diagrams to encode transformations.
result All knots can be transformed into a trefoil.
In mathematics, a knot is a single strand of string crossed over itself any number of times, and connected at the ends. The Reidemeister Moves have been proven to be the three core moves necessary to fully untangle a knot. Some knots can be untangled to a loop (the unknot), while others are fundamentally knotted. We de…
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.
Knot theory applied to proteins, distinguishing folded linear chains.
problem Classifying proteins as unknots when intra-chain interactions are ignored.
method Developing knot theory for folded linear molecular chains, considering self-bonding, and using Gauss codes and quandles.
result Extended knot theory to distinguish topologies of proteins with intra-chain bonds.
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 …
A new knot measure, the untwisting number, equals the algebraic unknotting number.
problem Measuring the minimum number of twists needed to untangle a knot.
method Defined and compared the untwisting number with the unknotting number.
result The algebraic untwisting number equals the algebraic unknotting number.
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 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.