New TQFT for link cobordisms without decoration.
problem Tackles the problem of defining a TQFT for link cobordisms.
method Introduces a Heegaard-Floer homology functor.
result Independently constructs a TQFT for link cobordisms.
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48 results for “decorated links”
The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how to find the 2-variable Homfly invariant <λ…
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 …
We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction, and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology.
The abstract extends Reidemeister theorem to 3-manifolds using diagrams of links and bands.
problem Extending Reidemeister theorem to 3-manifolds.
method Defines diagrams for links and bands on spines and flow-spines of 3-manifolds and describes combinatorial moves.
result Reproves and extends Brand et al. result on 3-manifolds.
Study geometric bases for A-polynomials in SU(3) using arcade formalism.
problem Understanding relations between different knots and their representations.
method Use arcade formalism and Kuberberg rule to simplify calculations.
result Equations for knot polynomials become independent of representation.
New equivalence relation on ribbon graphs connects to virtual links.
problem Understanding virtual links through ribbon graphs.
method Introducing a new equivalence relation on ribbon graphs.
result Correspondence between virtual links and ribbon graphs.
We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot…
We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the link homology categorifying the link polynomial. We also provide connections to the equivarian…
We construct {\it quantum hyperbolic invariants} (QHI) for triples , where is a compact closed oriented 3-manifold, is a flat principal bundle over with structural group $PSL(2,\mc)$, and is a non-empty link in . These invariants are based on the Faddeev-Kashaev's {\it quantum dilogarithms},…
Study essential diagrams of knots in and their relation to virtual knots.
problem Understanding essential diagrams and their relation to virtual knots in .
method Analyzing knots with minimal double lines and embedding virtual knot theory.
result Virtual knot theory is embedded in the theory of knots in .
We give an explicit construction of the Honda--Kazez--Matić gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link F…
Additional gradings on generalisations of Khovanov homology and invariants of embedded surfacesmath.GT
We define additional gradings on two generalisations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened surfaces and of surfaces embedded in thickened -manifolds. In particul…
Study of quantum decorated character stacks and their quantizations.
problem Quantization of decorated character stacks and their compatibility with cutting and gluing.
method Using stratified factorization homology, extend Fock and Goncharov's construction to include stacky points.
result Construction of categorical charts and flips on quantum decorated character stacks.
Study strip deformations of hyperbolic polygons with decorated vertices.
problem Understanding deformations of hyperbolic polygons with decorated vertices.
method Analyzing strip deformations of ideal hyperbolic polygons with horoballs.
result Arc complexes parameterize uniformly lengthening deformations.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
Decorated TQFTs compute invariants with additional structures.
problem Computing topological invariants with additional structures.
method Cutting and gluing to obtain decorated invariants, proposing Hilbert spaces for two-dimensional surfaces.
result Proposal for Hilbert spaces assigned to surfaces in decorated TQFTs.
Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
problem Understanding the structure of decorated hyperbolic polygons.
method Combinatorial approach using pseudo-manifolds and shellability.
result Arc complexes of decorated hyperbolic polygons are closed piecewise linear balls.
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group and a factorizable ribbon Hopf -bialgebra , constructs a TQFT for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in .
result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
Study compares constrained and decoupled moduli spaces of manifolds with particles and discs.
problem Comparing constrained and decoupled moduli spaces of manifolds with embedded particles and discs.
method Generalized Bödigheimer--Tillmann's work to higher dimensions and different tangential structures.
result New results for surfaces with different tangential structures and higher dimensional manifolds.
We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of π-algebraically-split links in 3-manifolds with fundamental group π. Using this class of links, we define a theory of finite type invariants of 3-manifolds in such …
Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
Proves conjecture linking cluster algebras and skein algebras for surfaces with punctures.
problem Cluster and skein algebras on surfaces with punctures.
method Geometric and algebraic methods, including decorated Teichmüller spaces and skein algebras.
result Cluster and skein algebras coincide for surfaces with at least 2 punctures.
Study q-series for 3-manifolds with line defects, proving homomorphism and conjecturing holomorphic modularity.
problem Understanding BPS -series for 3-manifolds with line defects.
method Proving homomorphism from skein module to space of -series, conjecturing holomorphic modularity.
result Holomorphic quantum modularity of -series suggests new approach to Langlands duality.
The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.
problem Understanding the geometric structure of decorated hyperbolic surfaces.
method Developing a characterisation of canonical tessellations and dual decompositions using hyperbolic geometry.
result Decorations on hyperbolic surfaces induce unique canonical tessellations and dual decompositions.
Corrects a 1-off error in Harer's spine dimension calculation for decorated Teichmüller spaces.
problem Incorrect dimension calculation in Harer's spine for decorated Teichmüller spaces.
method Identifies and corrects the dimension discrepancy in Harer's spine construction.
result Corrects the dimension of Harer's spine by 1 for decorated Teichmüller spaces.
This is a survey on the project `Decorated Marked Surfaces', where we introduce the decoration on a marked surfaces , to study Calabi-Yau-2 (cluster) categories, Calabi-Yau-3 (Fukaya) categories, braid groups for quivers with potential, quadratic differentials and stability conditions.
The paper constructs semistrict monoidal 2-categories from foam evaluations.
problem Creating examples of semistrict monoidal 2-categories.
method Using a closed foam evaluation formula as input, the paper rigorously constructs semistrict monoidal 2-categories.
result The constructed monoidal 2-categories are semistrict, have duals and adjoints, and carry a spatial duality structure.
DecoR estimates causal effects in confounded time series data.
problem Estimating causal effects in time series with unobserved confounders.
method Robust regression in the frequency domain.
result Proves upper bounds for estimation error of DecoR, implying consistency.
Enhanced Teichmüller space for surfaces with decorations and enhancements.
problem Parameterizing and understanding Teichmüller spaces with enhancements and decorations.
method Introduced a new variation of Teichmüller space, constructed parameterization, and introduced lamination space.
result Compatibility of shear coordinates and λ-length coordinates in the new deformation space.
Unified framework for smooth structures on coadjoint orbits.
problem Smooth structures on coadjoint orbits of diffeomorphism groups.
method Decorated and augmented nonlinear Grassmannians, functors, smooth structure.
result Uniform description of coadjoint orbits' smooth structures.
Study of decorated surfaces with vortices and their group structures.
problem Understanding group structures of decorated surfaces with vortices.
method Proved isomorphism between cluster braid group, braid twist group, and fundamental group of moduli space.
result Finite presentations of isomorphic groups were given.
Innovative series invariant for knot complements, linking to existing invariants.
problem Developing a new series invariant for knot complements.
method Introducing a three-variable series for plumbed knot complements.
result Deriving a surgery formula relating to invariant.
Discrete conformal maps on surfaces with vertex decorations are studied.
problem Discrete conformal equivalence for decorated piecewise Euclidean surfaces.
method Intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces, and convex hyperbolic polyhedra; concave variational principle.
result Proof of discrete uniformization theorem for decorated PE-surfaces.
We introduce a family of matrix dilogarithms, which are automorphisms of C^N tensor C^N, N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 -->…
We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving strands over and under traces and identify a new stop condition for the recursive…
We produce a one-parameter family of coordinates of the decorated Teichmüller space of an ideally triangulated punctured surface with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If , the decorated Teichmüller space in…
The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In t…
We are interested in the 3-Calabi-Yau categories arising from quivers with potential associated to a triangulated marked surface (without punctures). We prove that the spherical twist group ST of is isomorphic to a subgroup (generated by braid twists) of the mapping class group …
It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured F…
Let S be a path-connected, locally-compact CW-complex, and let M be a subcomplex with finitely-many components. A `decorated SL_2(C)-local system' is an SL_2(C)-local system on S, together with a choice of `decoration' at each component of M (a section of the stalk of an associated vector bundle). We study the (decorat…
The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptio…
We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the…
Develops a TQFT framework to compute invariants of three-manifolds.
problem Understanding the TQFT structure of invariants of three-manifolds.
method Decorated Spin-TQFTs, novel quantization of Chern-Simons theory, and algebra of observables.
result Explicit closed-form expressions for invariants of various three-manifolds.
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of puncture…
The punctured solenoid is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of . Furthermore…
Defines braids with double lines for links in a surface times circle and connects it to the affine Hecke algebra.
problem Presenting links in a surface times circle using braids with double lines.
method Defines braids with double lines, proves Alexander and Markov theorems, and connects Hecke algebra to affine Hecke algebra.
result The Hecke algebra of braids with double lines is isomorphic to the affine Hecke algebra.