Quadratic bounds found for graph dimensions.
problem Understanding dimensions of arc and disk graphs.
method Quadratic upper bounds calculation.
result Asymptotic dimensions of arc and disk graphs have been bounded.
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48 results for “arc graph”
Formula calculates distance between triangulations using arc graphs.
problem Calculating distances between triangulations efficiently.
method Proved a formula using projections into arc graphs.
result Distance formula for flip graph between triangulations.
Study shows arc and curve graphs are retracts of free group splitting complexes.
problem Understanding free group splittings through arc and curve graphs.
method Proved arc and curve graphs are coarse Lipschitz retracts of free splitting complexes and other related graphs.
result Arc and curve graphs are retracts of free group splitting complexes.
We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.
The grand arc graph's asymptotic dimension is shown to be infinite.
problem Determining the asymptotic dimension of the grand arc graph.
method Using Gromov-hyperbolic and cocompact arc and curve models, the asymptotic dimension is shown to be infinite for a broad class of surfaces.
result The asymptotic dimension of the grand arc graph is infinite.
Graph conditions ensure matching arc complexes are connected and hyperbolic.
problem Conditions for connectedness and hyperbolicity of matching arc complexes.
method Conditions on finite simplicial graphs guaranteeing connectedness and hyperbolicity of matching arc complexes.
result Conditions on finite simplicial graphs ensure connectedness and hyperbolicity of matching arc complexes.
Study of arc and curve graphs for surfaces of infinite type.
problem Understanding arc and curve graphs for surfaces with infinite topological type.
method Definition and analysis of arc and curve graphs for surfaces with a finite number of punctures, and study of subgraphs within curve graphs.
result Arc and curve graphs have infinite diameter and geometric rank 3, and are not hyperbolic.
Arc index of spatial graphsmath.GT
New upper bound found for arc index of spatial graphs.
problem Finding an upper limit for the arc index of spatial graphs.
method Extended the definition of arc presentation to spatial graphs and derived a new upper bound.
result The upper bound on the arc index of any spatial graph is lowest possible.
The paper studies geometric embeddings of arc graphs and their rigidity.
problem Investigating rigidity and convexity in geometric simplicial embeddings of arc-type graphs.
method Examining multiarc graphs and their rigidity properties under certain complexity conditions.
result Simplicial maps between certain multiarc graphs only arise in the 'obvious way' under necessary complexity conditions.
The flip graph and arc complex of a surface are shown to have finite rigidity.
problem Finite rigidity of flip graph and arc complex for surfaces.
method Embedding the flip graph in the arc complex and leveraging finite rigidity of the flip graph.
result Finite rigidity of the flip graph implies finite rigidity of the arc complex.
Generates special homeomorphisms for complex surfaces.
problem Creating specific homeomorphisms for infinite-type surfaces.
method General conditions for producing endperiodic loxodromics.
result Produces homeomorphisms acting loxodromically on arc graphs.
Shifts are not type-preserving on surface graphs.
problem Understanding the type-preserving property of shift maps on surface graphs.
method Analyzing Dehn twists and shift maps on arc, curve, and relative arc graphs of surfaces.
result Shift maps are not type-preserving on surfaces with isolated punctures.
Researchers describe the Gromov boundary of a graph related to surfaces.
problem Understanding the Gromov boundary of a graph associated with surfaces.
method Described a dense subset of the Gromov boundary as geodesic laminations, proving the graph satisfies a bounded geodesic image theorem.
result The boundary is not compact.
Prescribed Arc Graphsmath.GT
Defined a new graph type for compact surfaces, proving its connectedness and infinite diameter.
problem Understanding the structure of arc graphs on compact surfaces.
method Defining and analyzing the prescribed arc graph for compact surfaces with boundary and relations .
result The prescribed arc graph is connected and infinite-diameter, with specific conditions for Gromov hyperbolicity.
Study arcs on surfaces, focusing on topological aspects and group actions.
problem Understanding arcs and their complements on surfaces.
method Characterize infinite-type surfaces via homeomorphic subsurfaces, construct actions on arc graphs.
result New characterisation of infinite-type surfaces and actions on arc graphs.
The grand arc graphmath.GT
New simplicial complex for infinite-type surfaces shows graph properties.
problem Characterizing infinite-type surfaces using graph theory.
method Constructing grand arc graph and analyzing its properties.
result Grand arc graph is infinite-diameter and δ-hyperbolic under certain conditions.
Maps from surfaces to handlebodies are projected uniformly.
problem Mapping homotopy classes from surfaces to handlebodies.
method Uniformly Lipschitz retraction of sphere graph to arc graph.
result Retraction is uniformly bounded and close to nearest point projection.
Meander diagrams of knots and spatial graphs: proofs of generalized Jablan--Radović conjecturesmath.GT
Proves generalized meander conjectures for knots and spatial graphs.
problem Proving generalized meander conjectures for knots and spatial graphs.
method Study decomposition into simple arcs for diagrams of knots and spatial graphs.
result Proves generalized Jablan--Radović conjectures for knots and spatial graphs.
New proof for knot state-sum formula using bijection between states.
problem Proving a knot state-sum formula for colored Jones polynomial.
method Established bijection between states on arc-graph and bichromatic digraph, used flow property of R-matrix.
result Two state models are essentially the same, extending formula to links.
We extend the notion of unicorn paths between two arcs introduced by Hensel, Przytycki and Webb to the case where we replace one arc with a geodesic asymptotic to a lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the ar…
Hensel-Przytycki-Webb proved that all curve graphs of orientable surfaces are 17-hyperbolic. In this paper, we show that curve graphs of non-orientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb's argument. We also show that arc graphs of non-orientable surfaces are 7-hyperbolic, and arc-curve graphs …
Affine equivalence of half-translation surfaces via saddle connection graphs.
problem Understanding affine equivalence of half-translation surfaces.
method Association of saddle connection graphs and investigation of their automorphism groups.
result Every isomorphism between saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces.
Paper proves bounds on knot indices using bisected vertex leveling of plane graphs.
problem Upper bounds on knot indices using plane graph embeddings.
method Bisected vertex leveling of plane graphs to prove upper bounds on braid index and arc index.
result Quadratic upper bound on minimal crossing number of delta diagrams.
Paper tackles natural science exam questions, improving over previous systems.
problem Hard natural science exam questions requiring advanced logic reasoning.
method Constructs contextual knowledge graphs for questions and supporting sentences, learns to reason with neural embeddings.
result Model outperforms previous state-of-the-art QA systems on the ARC Challenge Set.
Paper discusses hyperbolic structures for Artin-Tits groups.
problem No specific problem stated, focuses on hyperbolic structures.
method Algebraic analogues of previously known structures on Artin braid groups.
result Presented several candidates for hyperbolic structures.
Study of infinite-type surfaces' automorphisms and graph structures.
problem Understanding automorphisms of infinite-type surfaces.
method Isomorphic mappings between extended mapping class groups and graph automorphism groups.
result Extended mapping class groups are isomorphic to graph automorphism groups.
For a given boundary set consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface spanning the boundary.
Study properties of self-similar continua with finite intersection property.
problem Characterize self-similar continua with finite intersection property.
method Prove intersection graph criterion, finite order theorem, and parameter matching theorem.
result All Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
Study of graphs from hexagon decompositions of surfaces.
problem Understanding geometric properties of hexagon decompositions.
method Define and analyze graphs associated with hexagon decompositions of surfaces.
result Quasi-isometric relationships between studied graphs and known groups.
Study the geometry of graphs on surfaces with infinitely-generated groups.
problem Understanding the large-scale geometry of subgraphs of arc and curve complexes on infinite-type surfaces.
method Describe the geometry of subgraphs of the arc and curve complexes of infinite-type surfaces, invariant under mapping class groups.
result Recover and extend results on the geometry of these complexes.
New findings on strong convexity in triangulations of convex polygons.
problem Understanding the structure of triangulations and their distances.
method Analyzing geodesic paths and flag triangulations in flip-graphs.
result Strong convexity properties of triangulations in convex polygons are not always preserved.
Polygonalisation complex models mapping class group, with geometric insights.
problem Geometric model for mapping class group.
method Cube complex construction, flip graph analysis, hyperplane families, crossing graph study.
result Generic surfaces have distinct polygonalisation complexes, with quasi-isometric crossing graphs.
New infinite-type loxodromic elements found in surface mapping classes.
problem Identifying infinite-type loxodromic elements in mapping classes of surfaces.
method Constructing infinite families of mapping classes acting loxodromically on the relative arc graph.
result Explicit construction and characterization of infinite-type loxodromic elements.
As an extension of the class of algebraic links, A'Campo, Gibson, and Ishikawa constructed links associated to immersed arcs and trees in a two-dimensional disk. By extending their arguments, we construct links associated to immersed graphs in a disk, and show that such links are quasipositive.
Defines a new family of curves in space with applications.
problem Finding shapes similar to whirls in space.
method Intrinsic equation of curvature and torsion, position vector with arc length parameter.
result Necessary and sufficient conditions for the existence of the family of curves.
Study shows saddle connection graph's geometry and quasi-isometry properties.
problem Characterize the geometry and quasi-isometry of saddle connection graphs.
method Proved 4-hyperbolicity and uniform quasi-isometry to a tree, used generalised unicorn paths.
result Saddle connection graph is not quasi-isometrically rigid and its boundary is straight foliations.
Bayesian structure learning is the NP-hard problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian structure learning under graph theoretic restrictions on the super-structure. The super-structure …
This paper restricts efficient geodesics to non-separating curves.
problem Finding efficient geodesics in the complex of curves.
method Analysis of the dot graph and surgeries.
result Efficient geodesics can be restricted to the non-separating curve complex.
Study proves hyperfiniteness of mapping class group actions on surface graphs.
problem Hyperfiniteness of mapping class group actions on surface graphs.
method Infinite unicorn paths and Gromov boundaries of arc and curve graphs.
result Proves hyperfiniteness of orbit equivalence relations induced by mapping class group actions.
New moves transform any virtual knot to a trivial knot.
problem Transforming virtual knots to trivial knots.
method Introducing arc shift and region arc shift moves.
result Any virtual knot can be transformed into a trivial knot using these moves.
This paper shows neural networks can solve complex graph problems efficiently.
problem Solving exact maximum flow computation and minimum spanning tree problems.
method Introduces Max-Affine Arithmetic Programs and shows equivalence to neural networks.
result Two combinatorial optimization problems can be solved with polynomial-size neural networks.
Defines knotting probability for spatial arcs.
problem Calculating the probability of knotting in spatial arcs.
method Projection of spatial arcs to planes, defining knotting probability for diagrams.
result Knotting probability defined for every oriented spatial arc.
New graphs show hierarchical hyperbolic properties, extending previous work.
problem Characterizing hierarchically hyperbolic properties of multiarc and curve graphs.
method Analyzing the geometric intersection number and using PMod(S) action.
result Multiarc and curve graphs are hierarchically hyperbolic.
Coloring curves on surfacesmath.GT
Study on coloring curves on surfaces, showing growth and unique colorability.
problem Chromatic number of curve graphs on surfaces.
method Analyzing separating curves and homology classes, using Kneser graphs and hyperbolic geometry.
result Chromatic number grows like k log k for separating curves, and uniquely t-colorable for homology classes.
Study on unknotting twisted knots using arc shift and region arc shift moves.
problem Unknotting twisted knots and finding bounds for region arc shift number.
method Introduced arc shift move and region arc shift move for twisted knots.
result Found families of twisted knots with specific arc shift and region arc shift numbers.
Study of curves in Teichmüller discs and their geometry.
problem Understanding the geometry of curves in Teichmüller discs.
method Analysis of systole and cylinder sets, quasiconvexity, Hausdorff distance, nearest point projections.
result Sets of curves are quasiconvex and agree up to bounded distance.
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
Non-trivialization probability of arc system in 3D space
problem Defining and generalizing the knotting probability of an arc diagram in 3D space
method Transforming polygonal arcs in 3D space into unique arc diagrams
result Introducing and generalizing the Non-Trivialization probability (NT probability) for arc systems in 3D space