In 1983 Conway and Gordon proved that any embedding of the complete graph into contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recen…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,742 papers · 148 categories
Trend · papers per month
48 results for “Hamiltonian knots”
New theorem shows every integer can be represented by knot summation.
problem Understanding the summation of knot coefficients.
method Analyzing spatial complete graphs and Hamiltonian knots.
result Every integer can be represented by knot summation.
This paper shows how pseudo-Anosov flows represent stable Hamiltonian classes and limits the ways 3-manifolds can be obtained from knots.
problem Understanding the canonical representatives of stable Hamiltonian classes and their implications for 3-manifolds.
method Explains the analogy between pseudo-Anosov flows and stable Hamiltonian classes and generalizes an argument to limit the ways 3-manifolds can be obtained from knots.
result There are finitely many pseudo-Anosov flows admitting positive Birkhoff sections on any given rational homology 3-sphere, and any 3-manifold can be obtained in at most finitely many ways as surgery on a fibered hyperbolic knot in .
We describe which knots can be obtained as cycles in the canonical book representation of K_n, the complete graph on n vertices. We show that the canonical book representation of K_n contains a Hamiltonian cycle that is a composite knot if and only if n>11 and we show that when p and q are relatively prime, the (p,q) t…
The paper studies knot types of clean intersections in a 3D space.
problem Identifying knot types in clean intersections.
method Using compactly supported Hamiltonian isotopy and DGA maps.
result Constraints on knot types of intersections.
We state and prove a correct version of a theorem presented in an earlier paper.
Clean intersections of Lagrangian knots in 3D are impossible.
problem Prohibiting clean intersections of certain knots in 3D symplectic geometry.
method Symplectic field theory and algebraic constraints on augmentation varieties.
result No Hamiltonian diffeomorphism can cleanly intersect a specific type of knot's conormal bundle.
Researchers establish a connection between knot homology and Lie algebra actions.
problem Understanding the HOMFLY-PT homology of torus knots.
method Constructing an explicit isomorphism and computing tautological class actions.
result The tautological class action extends to Hamiltonian vector fields and differentials in spectral sequences.
In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, th…
We give a Conway-Gordon type formula for invariants of knots and links in a spatial complete four-partite graph in terms of the square of the linking number and the second coefficient of the Conway polynomial. As an application, we show that every rectilinear spatial contains a nontrivial Ha…
The altenating knots, links and twists projected on the S_2 sphere are identified with the phase Space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossing points, the edges, to the stable and unstable manifolds, connecting the saddles. Each facxe is then orient…
Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previou…
This is a sequel to the authors' article [BKO](arXiv:1901.02239). We consider a hyperbolic knot in a closed 3-manifold and the cotangent bundle of its complement . We equip with a hyperbolic metric and its cotangent bundle with the induced kinetic energy H…
The study examines knot probabilities in confined lattice polygons.
problem Determining the relative knotting probabilities in confined lattice knots.
method Used Monte Carlo algorithms to enumerate conformations of lattice knots in a confined volume.
result Relative knotting probabilities are small, with the model dominated by unknots.
This paper uses sheaf theory to constrain knot types in clean intersections.
problem Understanding constraints on knot types in clean intersections.
method Microlocal sheaf theory and 3-manifold theory.
result Existence of a surjective homomorphism preserving longitude and meridian.
The alternating knots, links and twists projected on the sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond to the stable and unstable manifolds connecting the saddles. Each face is then …
We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the…
Quantum Knots and Mosaicsquant-ph
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantu…
Study on knots and dynamics on three-sphere, linking bounds, and upper action bounds.
problem Existence and properties of Reeb orbits on three-sphere with torus knots.
method Knot filtered embedded contact homology and obstructions to exact symplectic cobordisms.
result Upper bound on mean action of periodic orbits and linking bounds for Reeb orbits.
Classifies doodles into prime and super prime types, describing them with doodle codes.
problem Classifying doodles into prime and super prime types.
method Using doodle codes to describe complementary regions and enumerate doodle diagrams.
result Super prime doodles have a Hamiltonian circuit.
Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 mo…
This is the first of a series of two articles where we construct a version of wrapped Fukaya category of the cotangent bundle of the knot complement of a compact 3-manifold , and do some calculation for the case of hyperbolic knots $K …
This work generalizes Hamiltonian mechanics using closed differential forms.
problem Hidden invariants in classical Hamiltonian mechanics.
method Establishes a novel correspondence between generalized Hamiltonian mechanics and multisymplectic geometry.
result Key theorems linking classical and generalized Hamiltonian systems.
This thesis extends Hamiltonian actions to multisymplectic geometry, classifying actions on spheres and constructing homotopy comomentum maps.
problem Extending Hamiltonian actions to multisymplectic geometry.
method Explicit constructions and concrete examples of homotopy comomentum maps.
result Complete classification of compact group actions on multisymplectic spheres and explicit construction of homotopy comomentum maps.
Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.
problem Existence and properties of non-vertical fully-Hamiltonian vector fields in almost symplectic manifolds with Lagrangian fibrations.
method Investigates vector fields in 2n-dimensional almost symplectic manifolds with Lagrangian fibrations, focusing on partially-Hamiltonian and fully-Hamiltonian vector fields.
result Non-vertical fully-Hamiltonian vector fields exist under certain genericity conditions and can be reduced to families of symplectic-Hamiltonian vector fields.
Holographic energy equals Hamiltonian energy.
problem Equating holographic and Hamiltonian energies.
method Relative holographic and Hamiltonian energy comparison.
result Holographic energy is identical to Hamiltonian energy.
Develops Hamiltonian Score Matching and Generative Flows for machine learning.
problem Estimating score functions and designing generative models.
method Introduces Hamiltonian velocity predictors (HVPs) for score matching and generative flows.
result Hamiltonian Generative Flows (HGFs) rival leading generative modeling techniques.
Sum of Hamiltonian manifoldsmath.SG
Summing Hamiltonian manifolds with a common submanifold.
problem Combining Hamiltonian manifolds with a shared submanifold.
method Establishing symplectic reduction and comparing Chern classes.
result Symplectic reduction of the sum agrees with the sum of reductions.
The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.
problem Investigating stability properties of Hamiltonian Poisson integrators.
method Examples of Lotka-Volterra dynamics and numerical investigations of a non-integrable system are used.
result The existence of a modified Hamiltonian is crucial for the stability of Hamiltonian Poisson integrators.
The paper studies co-Hamiltonian diffeomorphisms on compact cosymplectic manifolds.
problem Fix-point theory and co-Hamiltonian diffeomorphisms on compact cosymplectic manifolds.
method Fix-point theory, Arnold's conjecture, co-Hofer norms, topologies, approximations lemmas.
result Minimum number of fix points for co-Hamiltonian diffeomorphisms is at least 1.
New algorithms improve MCMC efficiency for complex distributions.
problem High variance and low effective sample size in MCMC samplers.
method Antithetic Riemannian Manifold and Quantum-Inspired Hamiltonian Monte Carlo.
result Improved effective sample size and variance reduction.
The paper explores deformations of quasi-Hamiltonian spaces to Hamiltonian spaces.
problem Deforming quasi-Hamiltonian spaces to Hamiltonian spaces.
method Introducing and proving examples of deformations, including Lie groups and conjugacy classes.
result Moduli space of flat G-connections deforms to T*G^r+g.
Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian defo…
New integrators preserve geometric structure in Hamiltonian systems.
problem Preserving geometric structure in Hamiltonian systems on Jacobi manifolds.
method Combining Poissonization and symplectic bi-realizations to construct structure-preserving integrators.
result Explicit construction and application of Jacobi Hamiltonian integrators.
The paper develops a theory linking Hamiltonian and quasi-Hamiltonian manifolds.
problem Understanding the deformation of Hamiltonian quasi-Poisson manifolds to Hamiltonian Poisson manifolds.
method Introduces a generalized Hamiltonian deformation theory and constructs a topological quantum field theory.
result Shows that the imploded cross section of the double $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$.
Gauss diagrams' properties can change with Hamiltonian cycle choice.
problem The impact of Hamiltonian cycle choice on Gauss diagrams.
method Examined realizable and unrealizable Gauss diagrams, and proved preservation of realizability under certain Hamiltonian cycle changes.
result Properties of Gauss diagrams can vary with Hamiltonian cycle choice.
We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of cohomogeneity one (which is the same as being multiplicity free of rank one). Here the group acting is a compact connected Lie group (simply connected in the quasi-Hamiltonian case). This work is a concretization of the more general classific…
Classifies compact multiplicity free quasi-Hamiltonian manifolds.
problem Classifying compact, multiplicity free, quasi-Hamiltonian manifolds.
method Symplectic reductions and Lie group analysis.
result Recover old and find new examples of these structures.
The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.
problem Extending graph signatures to Klein graphs and foams.
method Developed an analogy of Murasugi's bounds and used signatures to lower bound knot properties.
result Lower bounds on negative orbifold Euler characteristics and unknotting numbers.
Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
problem Accelerating convex optimization
method Hamiltonian dynamics
result Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
Arnold-Liouville systems cannot be bi-Hamiltonian generically.
problem The bi-Hamiltonian structure of Arnold-Liouville systems.
method Proving that a specific class of smooth functions is a meagre subset for the Fréchet topology, which implies Arnold-Liouville systems cannot be bi-Hamiltonian.
result Generically, Arnold-Liouville systems cannot be bi-Hamiltonian.
New method calculates volume-renormalized mass from Hamiltonian perspective.
problem Calculating volume-renormalized mass for asymptotically hyperbolic manifolds.
method Using Michel's mass invariants and a reduced Hamiltonian perspective, the volume-renormalized mass is deduced.
result The reduced Hamiltonian recovers the volume-renormalized mass and its variations.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
Hamiltonian Monte Carlo converges to target distributions under mild conditions.
problem Establishing convergence of Hamiltonian Monte Carlo algorithms.
method Analyzing convergence for Hamiltonian Monte Carlo under mild conditions.
result Outputs converge to target distributions under specified conditions.
The paper discusses reducing Hamiltonian systems by scaling and standard symmetries, leading to Kirillov Hamiltonian systems.
problem Reduction of symplectic Hamiltonian systems by scaling and standard symmetries.
method Proof of Kirillov Hamiltonian systems and equivalence of reductions.
result Equivalent Kirillov Hamiltonian systems from different reduction orders.
The purpose of this short note is to prove the uniqueness of Hamiltonian volume minimizing Lagrangian submanifolds which are Hamiltonian isotopic to RP^n in CP^n modulo isometric group actions.
Develops integrators for contact Hamiltonian systems preserving geometric structure.
problem Creating integrators for dissipative systems with geometric structure.
method Structure-preserving splitting framework based on exact-contact subflows.
result Local universality of contact splitting integrators.
Differentiable simulations control molecular Hamiltonians for desired outcomes.
problem Control and learning of molecular Hamiltonians for desired outcomes.
method Differentiable simulations to differentiate Hamiltonians with respect to target observables.
result Control and learning of molecular Hamiltonians for desired outcomes.