Foams describe singular cobordisms for Khovanov homology.
problem Describing singular cobordisms for Khovanov homology.
method Combinatorial description of foams as 2-categories.
result Topological realization of type D arc algebra.
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48 results for “type D arc algebra”
With any non necessarily orientable unpunctured marked surface (S,M) we associate a commutative algebra, called quasi-cluster algebra, equipped with a distinguished set of generators, called quasi-cluster variables, in bijection with the set of arcs and one-sided simple closed curves in (S,M). Quasi-cluster variables a…
We find a new algebra isomorphic to Khovanov's arc algebra in characteristic 2.
problem Understanding Khovanov's arc algebra in characteristic 2.
method We introduce a new algebra and show isomorphisms over a base ring of characteristic 2.
result Khovanov's arc algebra is isomorphic to over a base ring of characteristic 2.
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.
We study the arc complex of a surface with marked points in the interior and on the boundary. We prove that the isomorphism type of the arc complex determines the topology of the underlying surface, and that in all but a few cases every automorphism is induced by a homeomorphism of the surface. As an application we ded…
We provide a presentation of the Roger and Yang's Kauffman bracket arc algebra for the once-punctured torus and punctured spheres with three or fewer punctures.
Defines new -foams and their algebras, showing equivalence and applications.
problem Developing new algebraic structures for .
method Introducing parameter-dependent -foams and their associated web and arc algebras.
result All specializations of these algebras are equivalent, with applications in link and tangle invariants.
Injective homomorphism proves no zero divisors in Roger-Yang skein algebra.
problem Injectivity of Roger-Yang's homomorphism and zero divisors in skein algebra.
method Hyperbolic geometric considerations, ideal triangulation, normal arcs.
result Injective homomorphism proves no zero divisors in Roger-Yang skein algebra.
Arc Operads and Arc Algebrasmath.GT
Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a compactification due to Penner of a space closely related to Riemann's moduli space. Algebras over thes…
Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.
Relative cellular algebrasmath.RT
The paper extends cellular algebras with relative orderings and explores their properties.
problem Generalizing cellular algebras with different partial orderings.
method Classification and construction of simple modules, characterizations, and examples.
result Examples of relative cellular algebras that are not cellular.
The paper generalizes Thurston's earthquake map to cluster algebras of finite type.
problem Tackling Thurston's earthquake map in the context of cluster algebras of finite type.
method Introducing a cluster algebraic generalization of Thurston's earthquake map, defined by gluing exponential maps.
result Proves an analogue of the earthquake theorem for cluster algebras of finite type, showing the cluster earthquake map is a homeomorphism.
New 2-representations link spectral enhancements in link homology.
problem Spectral enhancements in link homology.
method Skew Howe duality, frames, and multifunctors.
result Spectral 2-representations of categorified quantum groups.
Counts arcs in surfaces, proving convergence of geodesic currents.
problem Counting arcs of the same type in compact surfaces and related geometries.
method Derives convergence of geodesic currents to prove arc counts.
result Proves convergence of geodesic currents, leading to arc counting results.
Classifies objects in graded skew-gentle algebras using geometric models.
problem Classifying indecomposable objects in the derived category of graded skew-gentle algebras.
method Introduces new geometric models (punctured marked surfaces and binary surfaces) to classify objects.
result Integrates geometric models to classify objects in the derived category of graded skew-gentle algebras.
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.
We define an associative algebra AS_h(S) generated by framed arcs and links over a punctured surface S which is a quantization of the Poisson algebra C(S) of arcs and curves on S. We then construct a Poisson algebra homomorphism from C(S) to the space of smooth functions on the decorated Teichmuller space endowed with …
This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -- with or without self-intersections -- we associate an element of (the fractio…
This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove th…
Study on real algebraic curves' moduli spaces using a new complex.
problem Understanding the topology of moduli spaces of real algebraic curves.
method Defined a new complex, the -complex, to encode intersection patterns.
result Showed that mapping class groups are virtual duality groups and deduced orbifold homotopy group results.
Diagrammatically describes algebra equivalent to perverse sheaves on isotropic Grassmannians.
problem Equivalence between algebra modules and perverse sheaves on isotropic Grassmannians.
method Uses a Koszul algebra and a folding procedure from a Khovanov arc algebra to establish equivalence.
result Category of -modules is equivalent to perverse sheaves on isotropic Grassmannians.
Study centers of generalized skein algebras, showing almost Azumaya properties.
problem Understanding the center of generalized skein algebras.
method Generalized skein algebra generated by loops and arcs, computed center, discussed implications.
result Center of Muller-Roger-Yang skein algebra is almost Azumaya.
Homotopy types of curve and arc complexes are studied.
problem Understanding the homotopy types of curve and arc complexes.
method Proving homotopy equivalence and contractibility of complexes.
result Fine curve complex is homotopy equivalent to curve complex, fine arc complex is contractible.
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.
Proves constant scalar curvature Kähler metrics are very general.
problem Existence of constant scalar curvature Kähler metrics on smooth polarized varieties.
method Combining uniform arc K-stability and algebraic properties in families.
result The constant scalar curvature Kähler locus is very general.
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.
Defines an arc metric on Teichmüller spaces of infinite type surfaces with boundary.
problem Defining a metric on Teichmüller spaces of surfaces with infinite type and boundary.
method Using Basmajian identity and geometric conditions, an asymmetric metric (arc metric) is defined.
result An arc metric is constructed on the quasiconformal Teichmüller space of certain infinite type surfaces.
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 paper describes topological properties of arcs and crossings in knot theory.
problem Understanding the topological nature of arcs and crossings in knot theory.
method Topological description of arcs and crossings as isotopy classes of probes, homotopy classes of diagram elements.
result Sets of arcs and crossings are fundamental for algebraic objects like quandles, partial ternary quasigroups, biquandloids, and crossoids.
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.
Blanchet introduced certain singular cobordisms to fix the functoriality of Khovanov homology. In this paper we introduce graded algebras consisting of such singular cobordisms à la Blanchet. As the main result we give algebraic versions of these algebras using the combinatorics of arc diagrams.
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
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.
New spectral sequence connects link homology to Hochschild homology.
problem Computing Hochschild homology of link invariants.
method Uses spectral sequence with -page from Khovanov homology of links in .
result Spectral sequence converges to Hochschild homology of bordered Floer invariants.
Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these b…
Isomorphism found between Floer homology and contact geometry.
problem Connecting Floer homology and contact geometry.
method Demonstrated isomorphism between strand algebra and contact category.
result Direct correspondence between Floer homology and contact geometry concepts.
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.
Infinite rank surface cluster algebras extend traditional concepts to surfaces with accumulation points.
problem Extending surface cluster algebras to infinite surfaces with accumulation points.
method Consider infinite mutation sequences and hyperbolic structures to define cluster variables as lambda lengths of arcs.
result Established transitivity of infinite mutation sequences on triangulations of infinite surfaces and provided expansion formulas for cluster variables.
Geometric models for Lie algebras from simple singularities.
problem Classifying simply-laced simple Lie algebras.
method Using polygonal wheels derived from Milnor fibers of simple singularities.
result Geometric root systems are isomorphic to Lie algebras.
Finite rigid sets in arc complexes help classify surfaces.
problem Classifying surfaces based on their arc complexes.
method Constructing finite rigid sets in arc complexes of surfaces.
result Isomorphic arc complexes imply homeomorphic surfaces.
New homotopy refinements for tangle invariants.
problem Stable homotopy refinements for tangle invariants.
method Refined Khovanov and Chen-Khovanov spectra.
result Induces refinements of platform algebras and invariants.
Uniform Closure Method and Bayes classifier perform similarly in classifying open knots.
problem Classifying knots in open macromolecular chains.
method Used the Bayes MAP classifier and compared it to the Uniform Closure Method.
result Both methods have comparable accuracy and positive predictive value.
The paper explores when specific knot operations simplify diagrams.
problem Understanding when arc crossing changes simplify knot diagrams.
method Examined two types of arc crossing changes on link diagrams and determined when they are unknotting operations.
result Any two crossing points in an alternating knot diagram are arc crossing change admissible.
Infinite type surfaces can be perfectly divided into triangles.
problem Triangulating surfaces of infinite type.
method Showed arcs can be completed into triangulations if they intersect curves a finite number of times.
result Any surface of infinite type admits an ideal triangulation.
Proves left-orderability of mapping class groups of infinite-type surfaces.
problem Left-orderability of mapping class groups of infinite-type surfaces.
method Inductive construction of a stable Alexander system and ideal arc systems.
result Proves left-orderability using carefully chosen exhaustion by finite-type subsurfaces.
Khovanov spectra for tanglesmath.GT
New homotopy refinements for tangle invariants defined.
problem Stable homotopy refinements for tangle invariants.
method Defined stable homotopy refinements of Khovanov's arc algebras and tangle invariants.
result Stable homotopy refinements of Khovanov's arc algebras and tangle invariants defined.
We provide an explicit, finite set of generators for the Kauffman bracket arc algebra defined by Roger and Yang.
Study on OI surfaces with unique geometric properties.
problem Characterizing and classifying ortho-integral surfaces.
method Analyzing geodesic arcs and cosh-length properties.
result Infinitely many commensurability classes of OI surfaces arise as topologies vary.