Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
New transformations preserve link isotopy, showing complexity differences.
problem Link isotopy preservation with complexity differences.
method Introducing multiflypes of rectangular diagrams of links.
result Two diagrams of same complexity not related by simpler moves.
Method detects intersections between ellipses for Borromean linking.
problem Detecting Borromean linking between ellipses.
method Transforming one ellipse to a unit circle, examining intersections.
result Efficiently determines Borromean linking between ellipses.
This note explains how to transform Heegaard diagrams into framed link diagrams.
problem No specific problem stated; transformation of diagrams is the focus.
method Explains a procedure to transform Heegaard diagrams into framed link diagrams.
result Demonstrates a method to transform Heegaard diagrams into framed link diagrams.
New sequences prove some link diagrams can't be transformed by specific moves.
problem Proving some link diagrams can't be transformed by specific sequences of Reidemeister moves.
method Proved the existence of I-generalized ordered sequences to create simple transformations.
result Some link diagrams cannot be transformed by sequences of moves that increase, then decrease crossings.
Link projections with the same circle arrangement can be transformed by specific moves.
problem Characterizing link projections based on their circle arrangements.
method Local moves to transform link projections and analyze their circle arrangements.
result Two link projections have the same circle arrangement if and only if they can be transformed into each other by certain local moves.
Links can be transformed into strongly quasipositive links.
problem Understanding the relationship between different types of links.
method Generalized Rudolph's algorithm to show topological concordance.
result Every link is topologically concordant to a strongly quasipositive link.
Relational data representations have become an increasingly important topic due to the recent proliferation of network datasets (e.g., social, biological, information networks) and a corresponding increase in the application of statistical relational learning (SRL) algorithms to these domains. In this article, we exami…
The study explores if a link can be transformed into another using specific diagram manipulations.
problem Can a link be obtained from another using crossing exchanges and smoothings?
method The problem is approached from a computational complexity perspective, focusing on specific link types.
result For certain types of links (torus links and twist knots), there is an algorithm to determine if a link can be transformed into another using crossing exchanges and smoothings in polynomial time.
The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.
problem Volume conjecture for Reshetikhin-Turaev invariants of 3-manifolds with links.
method Volume conjecture, hyperbolic cone metrics, discrete Fourier transforms, change-of-pair operations.
result Volume conjecture proven for specific cases, provides approach to solving Volume Conjecture for hyperbolic 3-manifolds.
TransGCN combines GCNs with transformation assumptions for better link prediction in KGs.
problem Link prediction in knowledge graphs for understanding graph structure.
method Unified GCN framework with simultaneous learning of entity and relation embeddings, using transformation assumptions.
result TransGCN outperforms state-of-the-art models on FB15K-237 and WN18RR.
Two link diagrams are link homotopic if one can be transformed into the other by a sequence of Reidemeister moves and self crossing changes. Milnor introduced invariants under link homotopy called μˉ. Nanophrases, introduced by Turaev, generalize links. In this paper, we extend the notion of link homotopy to nanop…
New geometric transformations link discrete and continuous curve motions.
problem Establishing a connection between discrete and continuous curve motions.
method Infinitesimal Darboux transformations of smooth curves.
result Alternate geometric interpretation for semi-discrete mKdV equation.
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.
The potential function of the optimistic limit of the colored Jones polynomial and the construction of the solution of the hyperbolicity equations were defined in the authors' previous articles. In this article, we define the Reidemeister transformations of the potential function and the solution by the changes of them…
Classifies colored links and spatial graphs up to colored link-homotopy.
problem Classifying colored links and spatial graphs up to colored link-homotopy.
method Using Habegger-Lin theory for colored string links, and extending to colored links and spatial graphs.
result Classification of colored links and spatial graphs up to colored link-homotopy.
New invariant stops certain types of geometric transformations.
problem Obstructing decomposable Lagrangian cobordisms.
method Defined an invariant for Legendrian links.
result Obstructs decomposable Lagrangian cobordisms.
Riley "defined" the Heckoid groups for 2-bridge links as Kleinian groups, with nontrivial torsion, generated by two parabolic transformations, and he constructed an infinite family of epimorphisms from 2-bridge link groups onto Heckoid groups. In this paper, we make Riley's definition explicit, and give a systematic co…
New method classifies 4-component link homotopy using claspers.
problem Classifying link-homotopy classes of 4-component links.
method Using Habiro's clasper theory to modify Levine's algebraic computations.
result More symmetrical and schematic classification of 4-component link homotopy.
Legendre transformations link related integrable hierarchies.
problem Understanding relationships between integrable hierarchies.
method Legendre-type transformations of generalized Frobenius manifolds.
result Linear reciprocal transformations link related hierarchies.
A new graph generation model uses Mallat's scattering transform.
problem Unclear mathematical properties and difficulty in training good generative models for graphs.
method Proposes a graph generation model using a Gaussianized graph scattering transform.
result Demonstrates state-of-the-art performance in link prediction and graph/signal generation.
We simplify neural networks to 3D to study their topological changes.
problem Understanding how neural network layers affect low-dimensional topological invariants.
method Limiting each layer to a width of 3D space, tracking changes in linking numbers.
result ResNets and transformers are equally powerful in changing linking numbers.
Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.
problem Finding rational cuspidal curves and Zariski pairs in weighted projective planes.
method Construct families of curves using Cremona transformations, compute fundamental groups, and use blow-up-down decompositions.
result Discover new examples of rational cuspidal curves and Zariski pairs in weighted projective planes.
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from …
We say that a link L1 is an s-major of a link L2 if any diagram of L1 can be transformed into a diagram of L2 by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knot…
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used to reduce the number of crossings in a link and virtual link. For known unknots,…
Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.
problem Linking numbers between geodesics in arithmetic hyperbolic 3-folds.
method Analyzing integral of Kudla--Millson theta series over Seifert surfaces.
result Series converges to genus 2 Siegel modular forms of weight 2.
A new transform links rotating calorons to solutions of a differential equation.
problem Existence and characterization of rotating calorons.
method Formulated a Nahm transform to relate rotating calorons to solutions of a delayed-differential equation.
result Existence of an eight-parameter family of rotating calorons with nontrivial holonomy.
We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We…
Classifies small links in an unmarked solid torus.
problem Classifying knots and links in an unmarked solid torus.
method Invariants and Dehn twists to detect and transform links.
result Classification of all non-split links up to 6 crossings.
We establish a link between Archimedes' method of integration for calculating areas, volumes and centers of mass of segments of parabolas and quadrics of revolution by factorization via the moments of a balance and an integration technique for a particular integrable system, namely Bianchi's Bäcklund transformation for…
Paper constructs motifs from planar tilings for DP weaves and polycatenanes.
problem Creating complex entangled structures from periodic tilings.
method Combinatorial methodology using polygonal link transformations.
result Predicting the type of motif from a given tiling and polygonal link method.
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further,…
Embeds Riemannian manifolds with Anosov flows, linking classical and new theorems.
problem Embedding Riemannian manifolds with Anosov flows.
method Isometric embedding into a closed Riemannian manifold with Anosov geodesic flow.
result Direct link between classical and new theorems.
Jones polynomial coefficients of almost alternating links are nontrivial and have alternating signs.
problem Understanding the Jones polynomial of almost alternating links.
method Formulas for Jones polynomial coefficients, crossing change analysis, and diagram comparison.
result Jones polynomial coefficients of almost alternating links alternate in sign and are nontrivial.
Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location of zeros of the Alexander polynomial of the fibered link complement implying a s…
Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
Algorithm constructs Grushko decomposition of certain groups.
problem Decomposing fundamental groups of graphs of free groups.
method Analyzing vertex links of CAT(0) square complexes.
result Transforms complex to one with strong connectivity vertex links.
The paper connects Legendrian links to cluster algebras via microlocal methods.
problem Understanding the relationship between Legendrian links and cluster algebras.
method Microlocal parallel transport of sheaf quantizations of Lagrangian fillings.
result Existence of quasi-cluster A-structures and cluster Poisson structures. Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.
problem Understanding Sp(n)-instantons on hyperkahler manifolds with conical singularities.
method Relating Sp(n)-instantons to deformed instantons and studying their properties on hyperkahler manifolds.
result Sp(n)-instantons on hyperkahler manifolds correspond to tri-contact instantons on the 3-Sasakian link.
Monads link instantons and bow solutions in complex geometry.
problem Linking instantons and bow solutions via monads.
method Generalized ADHM-Nahm transform and monads.
result Established one-to-one correspondence between instantons and bow solutions.
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.
A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g+1)-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformatio…
Generative Link Sequence Modeling predicts future links in evolving networks.
problem Predicting future links in networks with evolving structures.
method Sequence modeling framework with self-tokenization to capture temporal link formation patterns.
result GLSM achieves best performance on AUC metrics compared to existing methods.
Graph Convolutional Gaussian Processes predict missing links.
problem Link prediction in large graphs.
method Simplified graph convolutions and variational inducing point method.
result Consistent improvements over existing models and competitive performance.
New Poisson transforms link differential forms on homogeneous spaces to Riemannian symmetric spaces.
problem Linking differential forms on homogeneous spaces to Riemannian symmetric spaces.
method Construction of Poisson transforms using finite dimensional representations of reductive Lie groups.
result Explicit design of Poisson transforms compatible with BGG-complex for real hyperbolic space.
Explains a 2D color exchange invariant correspondence to 3D linking numbers.
problem Understanding color exchange invariants in 2D dynamics and their 3D geometric interpretation.
method Visualizes invariants as linking of lines on a special surface with Arf-Kervaire invariant one, and interprets it as an obstruction to continuous transformation.
result Interprets a 2D color exchange invariant as a 3D linking number, providing a topological explanation.