Analog of Kauffman bracket for non-orientable knots in thickened surface.
problem Defining an invariant for non-orientable knots in a non-orientable surface.
method Proposes an analog of the Kauffman bracket polynomial with modified sign rules.
result Polynomial is an isotopy invariant and independent of classical Kauffman for orientable covers.
This paper calculates the non-orientable 4-genus for knots with 10 crossings.
problem Determining the non-orientable 4-genus for knots with a specific number of crossings.
method Calculating the minimal first Betti number of non-orientable surfaces smoothly embedded in a 4-ball with boundary the knot.
result The non-orientable 4-genus for knots with 10 crossings has been calculated.
J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to th…
Researchers calculate the non-orientable 4-genus for knots with 8 or 9 crossings.
problem Determining the minimum non-orientable surface complexity for knots with specific crossing numbers.
method Computed the non-orientable 4-genus for knots with 8 or 9 crossings using surface embeddings in 4-ball.
result Computed the non-orientable 4-genus for all knots with 8 or 9 crossings, proving conjectures and providing new slicing number bounds.
Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.
problem Measuring the minimum genus of non-orientable surfaces bounded by torus knots.
method Computed bounds and provided a generalized formula for non-orientable 4-genus of torus knots.
result Computed bounds and a generalized formula for non-orientable 4-genus of torus knots.
Study on Legendrian knots and their non-orientable Lagrangian fillings.
problem Conditions for Legendrian knots to have non-orientable exact Lagrangian fillings.
method Developed combinatorial obstructions and classified fillability for various knot families.
result Completely determined decomposably non-orientable fillability for alternating and plus-adequate knots.
New invariant distinguishes singular knots and links.
problem Classifying singular knots and links.
method Using oriented singquandles and weight functions at crossings.
result Distinguishes singular granny knot from singular square knot.
New method finds non-orientable knotted surfaces in 4D.
problem Reducibility of knotted surfaces in 4D.
method Elementary obstruction to reducibility.
result Construction of stably irreducible non-orientable surfaces.
In the symplectization of standard contact 3-space, R×R3, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus 0. We show that any Legendrian knot has a non-or…
Study non-orientable 4-genus for 11-crossing non-alternating knots.
problem Computing the non-orientable 4-genus for specific knots.
method Survey tools and use various techniques to calculate the invariant.
result Calculate non-orientable 4-genus for 11-crossing non-alternating knots.
Paper introduces an invariant for knots in non-orientable manifolds, akin to Turaev's comultiplication.
problem Defining an invariant for pseudo-classical knots in non-orientable thickening of a non-orientable surface.
method Introducing an invariant Δ that is an analogue of Turaev comultiplication, defined in terms of homotopy classes of loops on the surface. result Analogous invariants to affine index polynomial for pseudo-classical knots in non-orientable manifolds.
Two complete knot invariants from diagrams, finite or infinite.
problem Classifying knots completely.
method Constructed two invariants from knot diagrams, finite or infinite.
result Finite set reveals knotting number.
Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.
problem Exploring differences in non-orientable 4-genus for periodic knots.
method Analyzed p-periodic knots, showing differences in equivariant and non-equivariant non-orientable 4-genus.
result Differences exist in non-equivariant and equivariant non-orientable 4-genus for periodic knots.
This study simplifies verification of invariants in oriented virtual knots.
problem Verifying invariants of oriented virtual knots is complex and time-consuming.
method Identifying a minimal generating set of oriented virtual Reidemeister moves.
result A four-element subset serves as a generating set for oriented virtual Reidemeister moves.
Researchers refine the non-orientable 4-genus of torus knots using Batson's surfaces.
problem Finding the minimum non-orientable 4-genus for torus knots. method Developed and analyzed Batson's non-orientable spanning surfaces in B4. result Batson's surfaces minimize the non-orientable 4-genus among certain surfaces. Proves Alexander theorem for oriented Thompson group.
problem Establishing representation of oriented knots and links.
method Analogy with classical Alexander theorem for knots.
result Every oriented knot/link can be represented by an element of the oriented Thompson group.
Study on Whitehead doubles and their sliceness properties.
problem Understanding sliceness of Whitehead doubles of knots.
method Survey of techniques to obstruct sliceness and improve bounds on non-orientable genus.
result Improved bounds on non-orientable 4 genus of Whitehead doubles and genus 1 non-orientable cobordisms to cable knots.
Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.
problem Conditions for unknotting oriented virtual knots and links.
method Local moves (Delta, Sharp, Pass) for oriented virtual links.
result Lower bounds for the unknotting numbers are proven and shown to be best possible.
Enumerates knots up to five crossings and describes moves between them.
problem Counting and classifying knots up to a specific number of crossings.
method Generated tables of minimal diagrams and derived moves between knots.
result Conjecture about a lower bound for the triple-crossing number based on Alexander polynomial.
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
New rational band moves simplify knot classification.
problem Classifying knots using rational moves.
method Introduced oriented rational band moves and proved their effectiveness.
result Knots that can be unlinkified by rational moves are rationally slice.
The writhe polynomial invariant is proven for virtual knots via shell moves.
problem Determining equivalence of oriented virtual knots using writhe polynomials.
method Introducing shell moves to prove equivalence of writhe polynomials.
result Two virtual knots are equivalent if they can be transformed by shell moves.
Characterizes Legendrian knots with exact Lagrangian fillings.
problem Identifying Legendrian knots with exact Lagrangian fillings.
method Characterization through exact orientable Lagrangian fillings.
result Underlying smooth knot types of fillable Legendrian 4-plats are positive.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
Enhances knot and link invariant using tribracket modules.
problem Counting invariants of oriented knots and links.
method Introduces tribracket modules and uses them to enhance the tribracket counting invariant.
result Shows the enhancement is proper and provides examples.
Study on factorizations of knot polynomials for up to 12 crossings.
problem Understanding factorizations of HOMFLY polynomials for knots and links.
method Computer analysis of knots up to 12 crossings; irreducibility criterion for 2-connected plane graphs.
result Found 17 non-trivial factorizations of knots with up to 12 crossings.
Study knots with large SL2(C) character varieties.
problem Knots with high-dimensional character varieties.
method Two diagrammatic constructions: split link diagrams and rational tangle replacements; braids and orientation-reversing involutions.
result Conjecture that all Turk's head knots Th(p,q) with p and q odd are X-large. Knots in 3-manifolds are equivalent if isotopic, except in special cases.
problem Understanding when knots in 3-manifolds are equivalent and isotopic.
method Analyzing prime, closed, oriented 3-manifolds and irreducible manifolds, and considering orientation-preserving mapping class groups and homeomorphisms.
result Knots in prime, closed, oriented 3-manifolds are isotopic if and only if the orientation preserving mapping class group is trivial.
Study finds new knots that bound Möbius bands in 4D space.
problem Identifying knots that bound Möbius bands in 4D.
method Large-scale computational search with knot invariant obstructions.
result New examples of knots with non-orientable 4-genus equal to 1.
It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev inva…
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.
Ribbonness proven for slice knots, solving an old question.
problem Proving ribbonness for slice knots.
method Showing a link bounds a ribbon surface that is a renewal embedding of a bounding surface.
result Every slice knot is a ribbon knot.
Defines knotting probability for spatial arcs.
problem Calculating the probability of knotting in spatial arcs.
method Projection of spatial arcs to planes, defining knotting probability for diagrams.
result Knotting probability defined for every oriented spatial arc.
Characterizes knots with warping sum ≤ 3 and provides some knots with warping sum 4.
problem Understanding the warping sum of knots and its constraints.
method Characterization and example provision of knots based on their warping sum.
result Knots with warping sum ≤ 3 are characterized, and some knots with warping sum 4 are identified.
This paper finds all prime alternating knots with minimal warping degree two.
problem Finding knots with minimal warping degree.
method Examined all prime alternating knots and determined those with minimal warping degree two.
result All prime alternating knots with minimal warping degree two were identified.
Study extends knot genus results to two-component alternating links.
problem Determining genera of two-component alternating links.
method Unified quantity from splice sequence and correction term.
result Unified quantity completely determines orientable and non-orientable genera.
Introduces virtual tribrackets for knot coloring.
problem Coloring regions in virtual knot diagrams.
method Algebraic structure (virtual tribrackets) for knot coloring.
result Invariant can distinguish certain virtual knots.
We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of c…
This paper extends danceability concept to twisted virtual knots.
problem Danceability of knot diagrams on non-orientable surfaces.
method Expanding danceability concept to twisted virtual knot diagrams.
result Danceability properties of twisted virtual knots.
A new quantum invariant for virtual knots and links.
problem Quantum invariants for virtual knots and links.
method Generalization of biquandle brackets to parity biquandles.
result The new invariant is stronger than classical biquandle brackets for virtual knots.
Extends knot polynomial to knotted 4-valent graphs.
problem Constructing an invariant for knotted 4-valent graphs.
method Graphical calculus and Reidemeister moves for 4-valent graphs.
result Extension of sl(n) polynomial to knotted 4-valent graphs. Research confirms non-trivial knots in S^3 do not admit purely cosmetic surgeries.
problem Non-trivial knots in S^3 do not admit purely cosmetic surgeries.
method Analysis of JSJ-structures to confirm Gordon's conjecture.
result Connected sums of knots do not admit purely cosmetic surgeries.
New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
New knot polynomials distinguish knot orientations without using knot groups.
problem Distinguishing knots based on their orientations without relying on knot groups.
method Constructing combinatorial 1-cocycles on moduli spaces of knots and cables, using Gauss diagram formulas and local parameterization.
result Polynomial invariants that can distinguish knot orientations.
Study on quandle coloring quivers for (p, 2)-torus knots and links.
problem Understanding quandle colorings of (p, 2)-torus knots and links.
method Introduced quandle coloring quivers and studied them for dihedral quandles.
result Characterized quandle coloring quivers for (p, 2)-torus knots and links.
Proof of orientable 3-manifolds parallelizability using knot theory.
problem Proving all orientable 3-manifolds are parallelizable.
method Using knot theory and relationships between tangent and normal bundles.
result Completion and modification of a proof of Stiefel's theorem.