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.
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.
We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus for all positive braid knots with up to 12 crossings.
We establish a number of results about smooth and topological concordance of knots in S1×S2. The winding number of a knot in S1×S2 is defined to be its class in H1(S1×S2;Z)≅Z. We show that there is a unique smooth concordance class of knots with winding number one. …
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.
We extend Howie's characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficient conditions for a knot to be toroidally alternating. We also give a topological characterization of almost-alternating knots whi…
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. 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…
We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1] and the action of the concordance group of knots in the 3-sphere that ties in local knots. We prove that the trivial fre…
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.
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.
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…
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 …
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.
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. 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.
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 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…
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.
We give a complete characterization of the topological slice status of odd 3-strand pretzel knots, proving that an odd 3-strand pretzel knot is topologically slice if and only if either it is ribbon or has trivial Alexander polynomial. (By work of [FS85], a nontrivial odd 3-strand pretzel knot K cannot both be ribbon…
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.
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.
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.
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.
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.
Data science enhances knot theory by analyzing invariant relations.
problem Understanding the complex relations between knot invariants.
method Topological data analysis applied to knot theory.
result New insights into long-standing conjectures about knots.
The paper proves a conjecture about satellite knots and their slice genus.
problem The topological slice genus of satellite knots and its bounds.
method Establishes the conjecture for a variant of the topological slice genus, the Z-slice genus.
result The topological slice genus of a satellite knot is bounded above by the sum of the slice genera of the knot and the pattern.
New findings show infinitely many knots cannot be smoothly round handle slices.
problem Understanding the smoothability of knots in 4-dimensional space.
method Analyzing the properties of knots under surgery conjectures and cobordism conjectures.
result Infinitely many knots fail to be smoothly round handle slices.
We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples…
The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e. Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. This part w…
Machine learning maps knots to embeddings, revealing topological invariants.
problem Learning topological invariance in knot theory.
method Contrastive and generative machine learning techniques, auto-regressive decoder Transformer network.
result Neural networks can map different knots to the same point in an embedding vector space.
The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.