This paper constructs wild knots from beaded necklaces using a Schottky group.
problem Creating wild knots from beaded necklaces and studying their properties.
method Using a Schottky group generated by inversions on spheres to construct wild knots.
result The constructed wild knots are fibered if the original knot is fibered.
A study on linking configurations of horoball necklaces in hyperbolic space.
problem Linking problem for horoball necklaces in hyperbolic 3-space.
method Analyzing configurations of 8-bead necklaces around two diameter-one spheres.
result All beads must be of diameter one, spheres are tangent, and each bead kisses at least one sphere.
The Kontsevich integral of a knot is a powerful invariant which takes values in an algebra of trivalent graphs with legs. Given a Lie algebra, the Kontsevich integral determines an invariant of knots (the so-called colored Jones function) with values in the symmetric algebra of the Lie algebra. Recently A. Kricker and …
Kricker constructed a knot invariant Z^{rat} valued in a space of Feynman diagrams with beads. When composed with the so called "hair" map H, it gives the Kontsevich integral of the knot. We introduce a new grading on diagrams with beads and use it to show that a non trivial element constructed from Vogel's zero diviso…
Study of polygon spaces, characterizing critical points of area function.
problem Characterizing critical points of area function in polygon spaces.
method Geometric characterization of critical points and calculation of Morse indices.
result Generalization of isoperimetric theorems for polygons in the plane.
Study on polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
problem Understanding polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
method Analyzing polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
result Results generalize previous work by Katada and study polynomiality and outer nature of these functors.
Develops Johnson-Morita theory for 3D handlebody groups.
problem Algebraic structure of 3D handlebody groups.
method Analogue of Johnson-Morita theory for handlebody groups.
result Associated graded embeds into a Lie algebra of special derivations.
In this note, we calculate the leading term of the rational lift of the Kontsevich integral, introduced by Garoufalidis and Kricker, on the boundary of an embedded grope of class 2n. We observe that it lies in the subspace spanned by connected diagrams of Euler degree 2n-2 which have a bead t-1 on a single edge. This p…
New invariants for surface-links using biquandle modules.
problem Defining invariants for surface-links that are not determined by Alexander ideals.
method Enhancing biquandle counting invariant with biquandle modules.
result New invariants not determined by Alexander ideals.
New basis confirms Thurston's conjecture and reveals knot configurations.
problem Understanding cluster algebras and their bases from surfaces.
method Topological construction of band basis and comparison with Kazhdan-Lusztig type basis.
result Common triangular basis matches band basis in quantum cluster algebras.
A neural network method estimates entropy production from system trajectories.
problem Estimating entropy production from system trajectories without detailed dynamics.
method Developed a neural estimator (NEEP) for entropy production (EP).
result NEEP rigorously proves to provide stochastic EP by optimizing an objective function.
This is a substantially revised version. The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Eule…
A new method for joint noise removal and trend estimation from sparse signals.
problem Jointly removing noise and estimating trends from sparse signals.
method PENDANTSS combines SOOT/SPOQ penalties with BEADS algorithm in a Trust-Region block alternating variable metric forward-backward approach.
result Outperforms comparable methods in deconvolving analytical chemistry signals.
Study counts geodesics on modular surface, linking to necklace counting.
problem Counting geodesics on modular surface with specific winding numbers.
method Asymptotic expansion, generating function analysis, correspondence to necklace counting.
result Obtained asymptotic growth rate of m low-lying geodesics in terms of word length.
We address the problem of analyzing sets of noisy time-varying signals that all report on the same process but confound straightforward analyses due to complex inter-signal heterogeneities and measurement artifacts. In particular we consider single-molecule experiments which indirectly measure the distinct steps in a b…
Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [Bar-Natan, Garoufalidis, Rozansky and Thurston, arXiv:q-alg/9703025] and [Deligne, letter to Bar-Natan, January 1996, http://www.ma.huji.ac.il/~drorbn/Deligne/], which give, respectively,…
Study on knotting in very long polymer chains, finding Poisson distribution for prime knot types.
problem Understanding knotting in very long polymer chains.
method Generated and analyzed 243−k polygons of size n=2k using tree data structure and pivot algorithm. Used new knot diagram simplification and invariant-free classification. result Number of prime summands of knot type K in a random n-gon is well described by a Poisson distribution. This research optimizes plate structures to reduce vibrations in vehicles and aircraft.
problem Minimizing structural vibrations in engineering systems for improved passenger comfort.
method Guided flow matching design optimization integrating generative flow matching and surrogate model.
result Generated plate designs with reduced vibrations compared to random search and other methods.
New method uses normalizing flows to improve force fields for coarse-grained molecular dynamics.
problem Lack of reference atomistic forces makes force matching infeasible for MLCG force fields.
method Introduces noise-based kernels adapted to low-data regimes using normalizing flows.
result Flow-based kernels reduce local distortions while preserving global accuracy.
Machine learning improves coarse-graining of molecular dynamics models.
problem Creating accurate coarse-grained models for molecular dynamics simulations.
method Reformulated coarse-graining as a supervised machine learning problem using statistical learning theory and deep learning (CGnets).
result CGnets can capture multi-body terms and all-atom explicit-solvent free energy surfaces with fewer coarse-grained beads.
In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky's conjecture was soon proven by the second aut…