Formula for τ-invariant of satellite knots derived from L-space satellite operators.
problem Calculating the τ-invariant of satellite knots using L-space satellite operators.
method Algorithm to compute knot Floer complexes and formula for τ-invariant.
result Formula recovers existing formulas and proves new properties of τ-invariant.
Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set o…
Formula for satellite operators using knot Floer homology.
problem Computing knot Floer homology for satellite knots.
method Using Heegaard Floer Dehn surgery formulas and formal knot Floer complexes.
result Formulas to compute knot Floer homology for satellite knots.
We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank s…
Study satellite operations on knot invariant θ, proving additivity and distinguishing knots.
problem Behavior of knot invariant θ under satellite operations.
method Proved additivity, introduced computational tool, verified conjecture for 2977 knots.
result Distinguished knots using invariant θ and verified conjecture for 2977 primes.
Satellite operations with winding number ≠ 1 are not homomorphisms.
problem Characterizing homomorphisms in satellite operations.
method Using d-invariants of branched covers and Torelli group properties. result Satellite operations with winding number ≠ 1 are not homomorphisms.
Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance sp…
New knots found with non-trivial Steenrod operations on Khovanov homology.
problem Identifying knots with non-trivial Steenrod operations on Khovanov homology.
method Examined prime, hyperbolic, and satellite knots using Steenrod operations.
result Found knots (prime, hyperbolic, satellite) with non-trivial Steenrod operations on Khovanov homology.
New examples show satellite operations can expand the concordance group in topological knot theory.
problem Understanding how satellite operations affect the concordance group in topological knot theory.
method Forming satellites of knots with a fixed pattern and analyzing the induced map on the concordance group.
result Similar examples of rank-expanding satellite operations exist in the topological locally flat concordance group.
New infinite-rank summand found in knot concordance group.
problem Finding an infinite-rank summand in the smooth knot concordance group.
method Using iterated satellite operations with the Mazur pattern.
result Existence of a topologically slice knot K whose iterated satellites span an infinite-rank summand. Satellite operators generate infinite rank subgroups in knot concordance.
problem Understanding the structure of the knot concordance group under satellite operations.
method Using amenable L2-signatures, we analyze the image of iterated satellite operators. result The iterated satellite operator generates infinite rank subgroups in the knot concordance group.
Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of smooth concordance classes of knots. We give examples of winding number one satellit…
Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are…
Study satellite operators expanding concordance groups and their applications.
problem Understanding satellite operators and their impact on concordance groups.
method Floer-theoretic methods to analyze satellite operators and their effects on concordance groups.
result Established a Floer-theoretic condition for infinite-rank images of companion knots under satellite operators.
We study the effect of satellite operations on the Upsilon invariant of Ozsvath-Stipsicz-Szabo. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set {D2i,1}i=1∞ is a basis for an infinite rank summand of the group of smooth concordance classes o…
The paper explores slice disks and their properties using satellite operations and knot Floer homology.
problem Tackles the conjecture about slice disks and their positive Whitehead doubles.
method Uses techniques from knot Floer homology, Seiberg-Witten theory, and Khovanov homology.
result Provides evidence for the conjecture and constructs exotic disks.
Formula for Heegaard Floer multicurves of double tangles from knot complements.
problem Computing Heegaard Floer multicurve invariants of double tangles.
method Using a simple formula derived from knot complements, comparing with Khovanov homology.
result New characterisation of L-space knots and first example of thin knot Floer homology.
Formula calculates linking numbers in knot theory.
problem Obtaining obstructions for satellite operations.
method Explicit formula for linking numbers in branched covers.
result Evaluation of obstructions in various cases.
Links can be transformed into many others using a specific operation.
problem Understanding the relationship between strongly quasipositive links and their concordance.
method Used a satellite operation with a slice knot to transform links.
result Strongly quasipositive links can be transformed into infinitely many other links.
Combinatorial approach to compute satellite knot invariants using graph theory.
problem Computing knot invariants for satellite knots using bordered Heegaard Floer homology.
method Construct weighted A∞-modules using decorated planar graphs and prove their isomorphism. result Combinatorial proof of A∞ structure relations for the constructed modules. New constructions in Legendrian embeddings space yield novel invariants.
problem Understanding higher-order homotopy in Legendrian embeddings.
method Introduced parametric satellite and connected-sum constructions.
result Constructed new infinite families of Legendrian embeddings.
The Conway knot can't be smoothly tied to any other knot an infinite number of times.
problem Understanding the concordance order of knots.
method Rasmussen invariant, satellite operations, null-homologous twists.
result The Conway knot has infinite concordance order.
New operations transform braids into P-fibered braids.
problem Transforming braids into P-fibered braids.
method Satellite operations and full twists.
result Any braid can be transformed into a P-fibered braid.
Bayesian deep learning predicts satellite collisions.
problem Space debris poses planetary risk.
method Bayesian deep learning with LSTM networks.
result Predicts conjunction event evolution with uncertainties.
Introduces slice knots and concordance, linking to exotic smooth structures.
problem Understanding slice knots and their equivalence through concordance.
method Explains connections and introduces filtrations, satellite operations.
result Connections between slice knots and exotic smooth structures on \(\mathbb{R}^4\).
In the coming years, the satellite broadband market will experience significant increases in the service demand, especially for the mobility sector, where demand is burstier. Many of the next generation of satellites will be equipped with numerous degrees of freedom in power and bandwidth allocation capabilities, makin…
A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in S3. This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use…
Given a knot K in S3, a question raised by Cappell and Shaneson asks if the meridional rank of K equals the bridge number of K. Using augmentations in knot contact homology we consider the persistence of equality between these two invariants under satellite operations on K with a braid pattern. In particular…
We exhibit a knot P in the solid torus, representing a generator of first homology, such that for any knot K in the 3-sphere, the satellite knot with pattern P and companion K is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-…
Study Lagrangian zigzag cobordisms for Legendrian knots, comparing to smooth concordance.
problem Understanding Legendrian knots through Lagrangian cobordisms.
method Defined an equivalence relation on Legendrian knots using interpolating zigzag Lagrangian cobordisms, studied metric monoid, and compared to other concordance types.
result Proved structural results on torsion and satellite operators in the Lagrangian zigzag concordance classes.
Space debris warnings follow a predictable pattern, allowing timely satellite maneuvers.
problem Estimating when fresh information about space debris will arrive.
method Statistical learning model of the message arrival process, specifically a Bayesian Poisson process.
result The average prediction error for the next message arrival time is smaller than baseline predictions.
The Mazur pattern acts by the identity up to topological concordance.
problem Understanding the action of the Mazur pattern up to topological concordance.
method Comparing satellite operators and using topological concordance properties.
result Evidence that the Mazur pattern acts by the identity up to topological concordance.
We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of forming certain satellites of winding number zero. We also define strongly cosmetic crossing changes and we discuss their behavior under the operation of inserting full twists in the strings of closed braids.
New Seifert surfaces in 4-ball differ even when pushed in.
problem Finding distinct Seifert surfaces in 4-ball.
method Using cobordism maps on Khovanov homology.
result Examples of Seifert surfaces not isotopic in 4-ball.
Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.
problem Understanding gradings in annular Khovanov homology.
method Infinite families of annular links, computations, satellite operations, link splitting spectral sequence.
result Existence of links with unbounded annular Khovanov gradings but bounded Floer gradings.
Let P(K) be a satellite knot where the pattern, P, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, K, is a non-trivial knot in S3. We prove that P(K) is an L-space knot if and only if K is an L-space knot and P is sufficiently positi…
New Milnor's invariant condition for topologically slice links.
problem Conditions for topologically slice links involving Milnor's invariant.
method Different Milnor's invariant condition to handle cases not covered by previous theorem.
result New sufficient conditions for topologically slice links.
Satellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invarian…
The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.
problem Investigating exotic surface links in 4-ball.
method Two constructions of Brunnian exotic surface links, using satellite operations and covering links.
result Provided constructions of exotic Brunnian surface links with varied properties.
Single stabilization is not always enough to make exotic surfaces isotopic.
problem Determining if a single stabilization is sufficient to make exotic surfaces isotopic.
method Study of stabilization distance with satellite operations using Floer theoretic techniques.
result Found examples of exotic disks in the four-ball with arbitrarily large stabilization distance.
In this paper, we construct two families of satellite constructions for Brunnian links, called the satellite sum and the satellite tie. An interesting fact is that by applying the satellite sum and the satellite tie constructions, we can build infinitely many new Brunnian links from any given Brunnian links. With the h…
Machine learning competition predicts spacecraft collision risks.
problem Predicting future collision risks between orbiting satellites.
method Machine learning models trained on satellite collision data.
result Models accurately predicted collision risks with high precision.
Satellite knots can be trivialized by a single band move.
problem Satellite knots and their trivialization.
method Infinite family of satellite knots and a single band move.
result No disjoint band unknotting exists for satellite knots.
This study analyzes satellite communication latency using a stochastic geometry model.
problem Latency analysis of LEO satellite relay communication systems.
method Stochastic geometry framework with spherical BPP models, suboptimal satellite relay selection strategy.
result Derives distance distributions and analytical expressions for transmission delays.
Satellite links of fully positive braids are characterized.
problem Characterizing satellite links of fully positive braids.
method Analyzing fully positive braids and their satellites.
result Satellite links of fully positive braids are characterized by specific conditions.
Satellite links with many twists have simpler companions.
problem Relationship between satellite and companion links' complexity.
method Constructing satellite links with multiple full twists and analyzing their companion links.
result Satellite links with many twists have simpler companion links.
Develops a new framework for integrating satellite allocations in small portfolios.
problem Feasibility constraints in small portfolios, not return predictability, are the primary concerns.
method A four-layer feasibility framework: physical, economic, structural, and epistemic.
result Closed-form feasibility bounds on satellite size, turnover, and breadth without return forecasts.
FedSpace optimizes ML training on satellites and ground stations.
problem Training machine learning models on satellites with limited bandwidth.
method Federated Learning framework that dynamically schedules model aggregation based on satellite orbits.
result Reduces training time by 1.7 days over state-of-the-art algorithms.