New satellite knots counter a conjecture about Lorenz knots.
problem A conjecture about satellite knots and Lorenz knots was disproven.
method Constructed infinitely many counterexamples of satellite knots that are not cables.
result The conjecture was amended and shown to hold for many Lorenz knots.
The paper disproves a conjecture about satellite maps not inducing homomorphisms.
problem Satellite maps and their impact on knot concordance groups.
method Casson-Gordon signatures and n-solvable filtration analysis. result Examples of satellite maps that act like homomorphisms but do not induce them.
Satellite operations have infinite rank in smooth concordance group.
problem Understanding satellite operations in the smooth concordance group.
method Reduction to winding number zero satellites and use of SO(3) gauge theory. result Provides a criterion for satellite operations to generate infinite rank subgroups.
Thin position for knots in the 3-sphere was introduced by Gabai and has been used in a variety of contexts. We conjecture an analogue to a theorem of Schubert and Schultens concerning the bridge number of satellite knots. For a satellite knot K, we use the companion torus T to provide a lower bound for w(K), proving th…
Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.
problem Properties of twisted Mazur pattern satellite knots.
method Use bordered Floer theory to analyze knots and calculate genus.
result Prove non-thinness of Qn(K) and calculate 3-genus in terms of n and K. 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.
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…
Paper disproves conjectures about hyperbolic knots.
problem Conjectures about hyperbolic knots and their properties.
method Analyzes and refutes several conjectures about hyperbolic knots.
result Proves the conjecture about hyperbolic knots contradicts other plausible conjectures.
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.
In the paper we prove the conjecture by Alexander Zupan that w(K)⩾n2w(J) where w denote the width and K and J are satellite knot and its companion with winding number n. Also we proved that for satellite knot with braid pattern, the equality holds.
Equivalent conjectures proved for group presentations.
problem Balanced presentations of the trivial group.
method Equivalence of cyclic and satellite versions, restrictive cancellative version, stabilizations consideration.
result Original conjecture equivalent to cyclic version.
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 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.
Formulas for tau and epsilon concordance invariants of braided satellite knots
problem tau and epsilon invariants of satellite knots
method tau and epsilon invariants of braided satellite knots
result tau and epsilon formulas for braided satellite knots
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.
The paper studies knots formed by twisting a circle around a base knot and conjectures a linear growth in crossing numbers.
problem Understanding the growth rate of crossing numbers in twist families of knots.
method Introduced the stable crossing number and used geometric wrapping and algebraic winding to establish conjectures.
result The crossing number of Kn grows like nη(η−1) as no∞ for coherent twist families. Satellite knots with (1,1)-patterns have their Floer homology computed using immersed curves.
problem Computing the knot Floer homology of satellite knots with specific patterns.
method Using genus-one bordered Heegaard diagrams and immersed curves to compute the knot Floer chain complexes.
result Derivation of a formula for the τ-invariant of patterns derived from two-bridge links and computation of satellite knots' τ-invariant.
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…
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.
A technique to calculate the colored Jones polynomials of satellite knots, illustrated by the Whitehead doubles of knots, is presented. Then we prove the volume conjecture for Whitehead doubles of a family of torus knots and show some interesting observations.
In this note, I will discuss a possible relation between the Mahler measure of the colored Jones polynomial and the volume conjecture. In particular, I will study the colored Jones polynomial of the figure-eight knot on the unit circle. I will also propose a method to prove the volume conjecture for satellites of the f…
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.
Partial proof of a conjecture about knot concordance maps.
problem Proving a conjecture about homomorphisms in knot concordance.
method Analyzing self-maps of the knot concordance group.
result Proved a map is not a homomorphism for certain winding numbers.
New insights into L-space surgeries on multi-component links and satellites.
problem Characterizing rational surgery slopes yielding L-spaces for multi-component links.
method Generalizing Hedden and Hom's results for cables, computing L for satellites by torus-links and algebraic links. result Explicit descriptions of L for rational surgeries on multi-component links and satellites. Regular sliceness implies once-stably decomposable sliceness in symplectizations.
problem Relationship between regular and decomposable Lagrangian cobordisms in symplectizations.
method Stabilization-free strategy and satellite operations.
result Regular sliceness implies once-stably decomposable sliceness.
We provide a diagrammatic criterion for semi-adequate links to be hyperbolic. We also give a conjectural description of the satellite structures of semi-adequate links. One application of our result is that the closures of sufficiently complicated positive braids are hyperbolic links.
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…
We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots code…
Study of knot surgeries and JSJ decompositions to tackle L-space conjecture.
problem Understanding JSJ decompositions and L-space knots through Dehn surgeries. method Slope detection techniques applied to toroidal 3-manifolds and rational surgeries.
result JSJ graphs of certain knot exteriors are rooted intervals, implying left-orderable fundamental groups.
The only knots that are tunnel number one and genus one are those that are already known: 2-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Munoz and by Morimoto-Sakuma. This confirms a conjecture first made by Goda and Teragaito.
Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-…
The paper connects knot volume to A-polynomial structure.
problem Understanding the relationship between knot volume and A-polynomial structure. method Examining satellite knots and their A-polynomials to conjecture a connection with hyperbolic volume. result The conjecture that knots with zero hyperbolic volume have A-polynomials with specific factor structure. New series invariant for knots and cables, with robustness and relations.
problem Computing series invariants for complex knots and cables.
method Explicit computation and analysis of satellite knots, including a cable of the figure eight knot.
result First example of a cable knot with more than ten crossings, demonstrating robustness and integrality.
New satellite constructions create infinite Brunnian links.
problem Creating new Brunnian links from existing ones.
method Satellite sum and satellite tie constructions.
result Every Brunnian link has a unique tree-arrow structure.
Investigates ropelength of complex knots and links.
problem Establishing ropelength for knots and links beyond tested crossing numbers.
method Investigated torus knots up to 1023 crossings, satellite knots up to 42 crossings, and used a model of repeated Hopf links.
result Found power-law scaling in T(p,p+1) and derived formulae for predicting crossing-ropelength relationships.
New patterns create satellite knots with special surgeries.
problem Creating knots with specific surgical properties.
method Sufficient conditions for satellite knots to have L-space surgeries.
result New infinite families of patterns for satellite L-space knots.
Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.
problem Prove properties of fundamental groups of toroidal 3-manifolds.
method Slope detection using left-orders, foliations, and Heegaard Floer homology.
result Toroidal integer homology spheres have left-orderable fundamental groups.
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.
M. Scharlemann has recently proved that any genus one tunnel number one knot is either a satellite or 2-bridge knot, as conjectured by H. Goda and M. Teragaito; all such knots admit a (1,1) decomposition. In this paper we give a classification of the family of (1,1) knots in S3 with crosscap number two (i.e., boundi…
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.
We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures. To produce these examples, we introduce a new description of cork twists and util…
New methods link Legendrian satellites to Lagrangian cobordisms.
problem Understanding relations between Legendrian and Lagrangian knots.
method Constructing Lagrangian concordances through satellite operations.
result Maximum Thurston-Bennequin number restricts Legendrian satellite Lagrangian sliceness.
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.
New proof for minimizing tunnel systems in satellite chain links.
problem Minimizing the tunnel number of satellite chain links.
method Proving the tunnel number is minimized for links with a specific number of components and bridge number.
result The result is sharp for satellite chain links over a 2-bridge knot.
The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.
problem Preserving the diffeomorphism type of satellite 2-knots under the Gluck twist.
method Using new descriptions of satellite 2-knots, the paper shows that the Gluck twist does not change the diffeomorphism type of certain satellite 2-knots in three ways.
result The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.