New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
Characterizes toroidally alternating knots topologically.
problem Defining and characterizing toroidally alternating knots.
method Extending Howie's characterization to toroidally alternating knots and providing necessary and sufficient conditions.
result Necessary and sufficient conditions for a knot to be toroidally alternating.
New findings on knots that are both topologically and rationally slice.
problem Understanding knots that are both topologically and rationally slice.
method Analyzing the concordance group of knots in S3. result There are infinitely many topologically slice knots that are strongly rationally slice but not slice.
Quantum physics model uses knot theory for fragile topology.
problem Modeling quantum physics' fragile topology.
method Knot theoretic algorithm.
result Quantum physics' fragile topology modeled.
New knots found that can be cut topologically but not smoothly.
problem Existence of (1,1)-knots that are topologically slice but not smoothly slice. method Proof of existence of infinitely many (1,1)-knots. result Infinitely many (1,1)-knots topologically slice but not smoothly slice. This paper studies how knots combine using Alexander Polynomials.
problem How knots combine and their determinants behave.
method Basic knot theory, Alexander Polynomials, and composition techniques.
result Generalized solution for knot determinants in compositions.
Study on knots in S1imesS2 with unique smooth concordance class for winding number 1.
problem Understanding concordance classes of knots in S1imesS2. method Defined winding number and used it to classify knots, demonstrated distinctions between smooth and topological categories.
result Unique smooth concordance class for winding number 1, infinitely many topological concordance classes for other winding numbers.
Positive braid knots with maximal signature have maximal 4-genus.
problem Determining knots with maximal 4-genus.
method Analyzing positive braid knots and their signature invariants.
result All positive braid knots with up to 12 crossings have maximal 4-genus.
Characterizes almost alternating knots from generalizations of alternating knots.
problem Characterizing almost alternating knots.
method Topological characterization based on generalizations of alternating knots.
result A topological characterization of almost alternating knots.
Characterizes topological sliceness of 3-strand pretzel knots.
problem Determining the topological sliceness of 3-strand pretzel knots.
method Computations of Casson-Gordon 3-manifold signature invariants for double branched covers.
result Odd 3-strand pretzel knots are topologically slice if and only if they are ribbon or have trivial Alexander polynomial.
Construct divide knots with specific genus properties.
problem Understanding the difference between smooth and topological four-genus for knots.
method Construct divide knots with controlled smooth and topological four-genus ratios.
result For strongly quasipositive fibred knots, the ratio between smooth and topological four-genus can be made arbitrarily close to zero.
Study shows differences between smooth and topological almost concordance of knots.
problem Disparity between smooth and topological almost concordance of knots.
method Defined and analyzed almost concordance in 3-manifolds, proving results through infinite families and specific examples.
result Infinite families of knots exist that are topologically concordant but not smoothly almost concordant.
Study confirms conjectures for topologically slice knots' concordance group.
problem Primary decomposition conjectures for knot concordance groups.
method Use of amenable L2-signatures, Ozsváth-Szabó d-invariants, and Némethi's Heegaard Floer homology results. result Smooth concordance group of topologically slice knots has large subgroup with true primary decomposition.
Invariants for colored links found, with topological protection of certain knots.
problem Finding invariants for colored links and understanding their topological stability.
method Equivariant bordism groups of Conner and Floyd, topological surgeries.
result Topological protection of certain tricolored knots, with specific transformations.
Classifies Legendrian knots with maximal Thurston-Bennequin number.
problem Classifying Legendrian knots of a specific type.
method Using Thurston-Bennequin number as a criterion.
result Confirms conjectures about Legendrian knots of type 76. Computational topology shows knots can converge to stick knots.
problem Understanding convergence of smooth knots to stick knots.
method Computational experiments and mathematical visualization.
result Sufficient conditions for isotopic equivalence of knots.
New method detects and compares folding pathways of knotted proteins.
problem Understanding the function of knots in protein folding.
method Topological analysis of protein knotoid distributions and entanglement.
result Reveals unique folding pathway for shallow knotted Carbonic Anhydrases.
Topological quantum computers use hyperbolic knots for computations.
problem The difficulty of calculating quantum invariants of knots.
method Using hyperbolic knots to compute topological quantum computer invariants.
result The hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.
We give infinitely many examples of 2-bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12-crossing knot 12a255. These also provide the first known examples of alternating knots for which the smooth and topological genera differ.
We construct an infinite family of smoothly slice knots that we prove are topologically doubly slice. Using the correction terms coming from Heegaard Floer homology, we show that none of these knots is smoothly doubly slice. We use these knots to show that the subgroup of the double concordance group consisting of smoo…
Introduces a Cost function to measure Legendrian knot obstructions.
problem Measuring obstructions for Legendrian knot isotopies.
method Introduces a non-negative integer-valued Cost function.
result Cost function induces a metric on topologically isotopic Legendrian knots.
Alternating knots can't represent certain concordances.
problem Understanding which knots can represent concordances.
method Analyzing properties of alternating knots and their concordances.
result No nontrivial alternating knot can represent certain concordances.
Study shows knots can be reversed without becoming trivially concordant.
problem Understanding when knots can be reversed without changing their concordance class.
method Constructed an infinite family of knots, analyzed their concordance properties.
result Found an infinitely generated free subgroup in the concordance group of topologically slice knots.
Researchers create topologically protected knots in a realizable system.
problem Creating topologically protected vortex knots in experimentally realizable systems.
method Investigated non-Abelian vortices in tetrahedral order in spin-2 Bose--Einstein condensates and bent-core nematic liquid crystals.
result Discovered the first topologically protected knots in an experimentally realizable system.
Study calculates fundamental groups of torus knots using algebraic topology.
problem Calculating the fundamental group of torus knots.
method Algebraic topology and group theory.
result Computed fundamental groups of torus knots.
Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on S3 and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on S3 and their invariants encoded in colored HOMFLY polynomials by means of topological recursion. In the context of the …
The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of Ozsvath and Szabo's Heegaard-Floer theory, we show the existence of an infinite sub…
Study various topologies on polynomial knots and their homotopy types.
problem Understanding different topologies on polynomial knots and their impact on homotopy types.
method Examined spaces of polynomial knots in Rn with various topologies and studied their homotopy types. result The homotopy type of the space of polynomial knots in R3 is S2. Classifies uncolored bonded knots with up to 7 singularity points.
problem Classifying uncolored bonded knots with up to 7 singularity points.
method Generation of planar graphs, conversion into bonded knot diagrams, use of Yamada polynomial, and brute-force Reidemeister moves.
result Systematic classification of uncolored bonded knots with singularity number at most seven.
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.
The paper uses twisting operations to bound knot genus.
problem Bounding the topological slice genus of knots.
method Develops null-homologous twisting operations to study algebraic genus.
result New upper bounds on algebraic genera of torus and satellite knots.
Study shows infinite rank in bipolar filtration of topologically slice knots.
problem Understanding deeper structures in the smooth concordance group of topologically slice knots.
method Used higher order amenable Cheeger-Gromov L2 ρ-invariants and infinitely many Heegaard Floer correction term d-invariants. result Graded quotient of bipolar filtration has infinite rank at each stage greater than one.
The study explores how surface diffeomorphisms of knots relate to their topological properties.
problem Understanding how properties of surface diffeomorphisms of knots relate to their topological properties.
method Examining both braid and fibered knot perspectives to explore the relationship between surface diffeomorphisms and knot properties.
result Properties of surface diffeomorphisms may relate to four-dimensional topological properties of knots, such as the slice genus.
New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
Survey of invariants for knotted 2-spheres in 4-space.
problem Characterizing knotted 2-spheres in 4-dimensional space.
method Algebraic topology of knot exterior, gauge theory, combinatorial methods.
result Details are scarce and new results inexistent.
New method confirms conjectures about specific Legendrian knots.
problem Classifying Legendrian knots of specific topological types.
method New method of comparing Legendrian knots with nontrivial symmetry group.
result Conjectures about specific Legendrian knots are confirmed.
The paper explores how topological methods can reveal insights into electric charge distributions on knots.
problem Understanding the qualitative behavior of electric potentials on knots.
method Geometric topology techniques applied to electrostatics.
result Proved a lower bound on the size of the critical set based on knot projections.
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In…
Study shows knots can have large genus difference from concordance.
problem Understanding genus differences in knots and surfaces.
method Analyzes the topological 4-genus and minimal genus of bounded surfaces.
result Arbitrarily large genus difference between knots and their concordance.
Study of knotted defects in smectic liquid crystals using topological knot theory.
problem Understanding the topological structure of knotted defects in smectic liquid crystals.
method Investigation of screw and edge dislocations, focusing on their radial surface structure and knot fibration.
result Established a connection between smectic defects and knot theory, revealing the topological knotting of defects.
Machine learning reveals hidden features in knot classification.
problem Classifying the topology of closed curves.
method Investigating shortcut methods used by ML for knot classification.
result Developed a dataset and code to remove non-topological features.
Simple condition proves when 2-bridge knots are quasipositive.
problem Characterizing quasipositive two-bridge knots.
method Continued fraction expansion of p/q to prove a necessary and sufficient condition.
result Simple condition proves when 2-bridge knots are quasipositive.
Characterizes alternating knot exteriors using special surfaces.
problem Understanding topological properties of alternating knots.
method Topological characterisation using special spanning surfaces and a normal surface algorithm.
result Alternating is a topological property of knot exteriors, not just diagrams.
Sharp signature bound for 3-strand torus knots proved.
problem Determining the topological 4-genus of 3-strand torus knots.
method Using McCoy's twisting method and improving upper bounds.
result Signature bound is sharp for 3-strand knots and off by at most 1 for 4- and 6-strand knots.
We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.
Study satellite operations on Upsilon invariant of knots.
problem Effect of satellite operations on Upsilon invariant of knots.
method Analyzes satellite operations on Upsilon invariant of Ozsvath-Stipsicz-Szabo.
result Shows independence of knots and their satellites in certain cases.
New algebraic structures for topological pairs.
problem No specific problem stated; generalization of knot quandles.
method Introducing multi-quandles for topological pairs.
result New algebraic structures for topological pairs.
Explains how knots relate to 4D shapes.
problem Understanding 4D shapes through knot theory.
method Combines knot theory with 4D manifold topology.
result Connects 4D shapes to knot theory and other geometries.