Enhances quantum invariants using tribracket brackets.
problem Quantum invariants of tribracket-colored knots and links.
method Introduces tribracket brackets as skein invariants.
result Provides new quantum invariants and examples.
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48 results for “tribracket brackets”
Introduces entropic tribrackets and their applications in link distinguishing.
problem Distinguishing links with the same counting invariant.
method Definition and study of entropic tribrackets and their homsets.
result Homsets of entropic tribrackets form new entropic tribrackets.
New tribrackets defined to count link homotopy invariants.
problem Counting invariants of link homotopy.
method Defined -tribrackets and showed their invariants.
result Counting invariants for certain tribrackets are trivial.
Tribracket Polynomialsmath.GT
Innovates polynomial invariant for tribrackets.
problem Counting and distinguishing knots and links.
method Introduces subtribracket polynomials and uses them to enhance counting.
result Enhanced counting invariant for knots and links.
Multi-tribracketsmath.GT
We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular we consider the case of component multi-tribrackets which have different tribracket operations at single-component crossings and multi-componen…
Tribracket Modulesmath.GT
Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modu…
Quantum polynomials are derived from a specific tribracket structure.
problem Quantum enhancement polynomials for oriented links.
method Defined using a canonical two-element tribracket, proving polynomials can be derived from five specific ones.
result Universal quantum enhancement polynomials are strictly stronger than the Jones polynomial.
We introduce a new algebraic structure called \textit{local biquandles} and show how colorings of oriented classical link diagrams and of broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles and show that it is isomorphic to Niebrzydowski's tribracket (co)hom…
Virtual Tribracketsmath.GT
We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular we show that the in…
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.
Psybrackets define invariants for complex knots and links.
problem Defining invariants for complex knots and links.
method Introduced algebraic structures called psybrackets and used them to define invariants of pseudoknots and singular knots and links.
result Examples and computations provided for the invariants defined.
Invariants for trivalent graphs using algebraic colorings.
problem Creating an invariant for virtual trivalent spatial graphs.
method Colorings by virtual Niebrzydowski algebras.
result Generalization and computational implementation of invariants.
New bracket unifies nonholonomic dynamics and Hamilton-Jacobi theory.
problem Unified description of nonholonomic dynamics and Hamilton-Jacobi theory.
method Defined and proved coincidence of three nonholonomic brackets.
result Three nonholonomic brackets coincide.
New bracket theory connects three nonholonomic dynamics models.
problem Nonholonomic dynamics and their bracket formulations.
method Definition and proof of equivalence of three nonholonomic brackets.
result Three nonholonomic brackets are equivalent.
Flat connections derived from Poisson brackets on loop spaces.
problem Understanding the structure of Poisson brackets on loop spaces.
method Defined connections by explicit linear combinations of standard connections associated with the Poisson bracket.
result Connections are shown to be flat.
It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…
The paper constructs compatible Poisson brackets on gl(N).
problem Constructing compatible Poisson brackets on gl(N).
method Using constant tensors and Schouten brackets, the paper explicitly constructs quadratic Poisson brackets compatible with the standard Lie-Poisson bracket.
result Explicit construction of quadratic Poisson brackets compatible with the standard Lie-Poisson bracket on gl(N).
Derived bracketsmath.DG
We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp…
Researchers attempt to categorify biquandle brackets using Khovanov homology methods.
problem Categorify biquandle brackets using Khovanov homology methods.
method Outline a Khovanov homology-style construction for biquandle brackets.
result A canonical biquandle 2-cocycle is defined, but not a true categorification of biquandle brackets.
Explains biquandle brackets and quivers for a topology talk.
problem None explicitly stated; focuses on background information.
method Review of biquandle concepts and related structures.
result Clarifies understanding of biquandle bracket quivers.
An inclusive curvature-like framework for describing dissipation: metriplectic 4-bracket dynamicsmath-ph
A new framework describes dissipation using a metriplectic 4-bracket.
problem Describing dissipation in a way that preserves energy and entropy.
method Using a metriplectic 4-bracket, a quantity like the Poisson bracket with symmetries motivated by Riemannian curvature.
result The metriplectic 4-bracket dynamics includes all known previous binary bracket theories for dissipation.
New biquandle bracket invariants are linked to biquandle 2-cocycles.
problem Quantum enhancements and biquandle colored links.
method Proving biquandle bracket invariants are pointwise products of other invariants and biquandle 2-cocycles.
result New biquandle bracket invariants are equivalent to the Jones polynomial on knots.
New geometric definition of Lie bracket for undirected curves.
problem Understanding the Lie bracket of undirected curves on a surface.
method Local geometric definition and proof of three results.
result The TWG bracket counts intersection and suggests disjoint representatives.
New examples show non-trivial parity-biquandle bracket.
problem Constructing non-trivial parity-biquandle bracket examples.
method Slightly changed notation and constructed examples of knots and links.
result Minimality theorem: graphs appear as link invariants.
Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In th…
The paper defines symmetric brackets for skew-symmetric algebroids with totally skew-symmetric torsion.
problem Defining symmetric brackets for skew-symmetric algebroids.
method Using connections with totally skew-symmetric torsion and pseudo-Riemannian metrics.
result Explicit formula for the Levi-Civita connection and symmetric brackets on almost Hermitian manifolds.
In this note we prove that, for a vector bundle over a manifold , a Dorfman bracket on anchored by and with a vector bundle over , is equivalent to a lift from to linear sections of , that intertwines the given Dorfman bracket w…
Extend Kauffman bracket skein module to homology theory using Heegaard splittings
problem Extend Kauffman bracket skein module to homology theory
method Combinatorial approach using Heegaard splittings
result Homology theory depends on Heegaard splittings
Introduces a new bracket for multicontact geometry and applies it to field theories.
problem Developing a new mathematical structure for multicontact geometry.
method Introducing a graded Jacobi bracket and multisymplectization.
result Established a new bracket that extends contact geometry concepts.
A new invariant for knotted graphs defined by label bracket.
problem Defining an invariant for knotted trivalent graphs.
method Generalizing Akimova and Manturov's construction to define the label bracket.
result The label bracket defines an isotopy invariant of knotted trivalent graphs.
We propose an extension of n-ary Nambu-Poisson bracket to superspace R^{n|m} and construct by means of superdeterminant a family of Nambu-Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace R^{n|m}. We prove in the case of …
Computes Kauffman bracket polynomial for specific 2-tangle shadows.
problem Calculating Kauffman bracket polynomial for complex tangle structures.
method Computed Kauffman bracket polynomial for specific 2-tangle shadows with up to 4 crossings.
result Computed polynomial for specific 2-tangle shadows.
New Poisson bracket connects to logarithmic manifolds.
problem Constructing a new Poisson bracket compatible with existing structures.
method Developed a new local Poisson bracket compatible with Adler-Gelfand-Dickey brackets, leading to a dispersionless limit.
result Leading term defines a logarithmic Dubrovin-Frobenius manifold.
In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure an…
Paper derives explicit formulas for AJ-bracket of tied links.
problem Lack of state-sum formula for AJ-bracket of tied links.
method Analyzed AJ-states of 2- and 3-tied link diagrams, derived resolution trees, and state-sum formulas.
result Derives first closed-form expressions for AJ-bracket.
Goldman bracket distinguishes surface homeomorphisms.
problem Characterizing homeomorphisms between non-compact surfaces.
method Using the Goldman bracket to distinguish homeomorphisms.
result A homotopy equivalence is a homeomorphism if it preserves the Goldman bracket.
The paper defines and proves equivalence of nonholonomic brackets in contact mechanical systems.
problem Nonholonomic constraints in contact geometry.
method Construct a general framework for non-holonomic constraints, define and prove equivalence of different nonholonomic brackets.
result All nonholonomic brackets coincide and one is an almost Jacobi bracket.
We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathem…
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…
A new double quasi-Poisson bracket on surface groups.
problem Constructing a new mathematical structure on surface groups.
method Proposing and proving a double quasi-Poisson bracket on group algebras.
result The double quasi-Poisson bracket is a noncommutative generalization of the Goldman bracket.
Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.
problem Quantizing the relationship between Koszul and Schouten brackets in Poisson geometry.
method Employing Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of -algebroids.
result Quantizes the -morphism into a single linear operator, a formal Fourier integral operator.
Biquandle Power Bracketsmath.GT
New invariants for link analysis include biquandle power brackets.
problem Analyzing oriented links with new invariants.
method Introducing biquandle power brackets as an infinite family of link invariants.
result Biquandle power brackets encompass classical and previous invariants.
Kaestner Bracketsmath.GT
A new quantum invariant for virtual knots and links.
problem Quantum invariants for virtual knots and links.
method Generalization of biquandle brackets to parity biquandles.
result The new invariant is stronger than classical biquandle brackets for virtual knots.
We derive a formula expanding the bracket with respect to a natural deformation parameter. The expansion is in terms of a two-variable polynomial algebra of diagram resolutions generated by basic operations involving the Goldman bracket. A functorial characterization of this algebra is given. Differentiability properti…
Paper disproves a theorem about Kauffman bracket skein module structure.
problem Disproving a 22-year-old theorem about Kauffman bracket skein module structure.
method Analyzing handle slidings on compressing discs in handlebodies.
result More relations found than previously predicted for connected sum of handlebodies.
Paper defines new invariants for surface-links using graph diagrams and magmas.
problem Tackles invariants for surface-links in entropic magmas.
method Uses marked graph diagrams and a generalization of Kauffman bracket magma.
result Defines new invariants for surface-links in 4-space.
For any n>1 we define an isotopy invariant, <Gamma>_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link…