Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
This study introduces a unified cohomology theory for braided algebras.
problem Classifying infinitesimal deformations of braided algebras.
method Developed a cohomology theory unifying Hochschild and Yang-Baxter cohomology.
result The second cohomology group classifies infinitesimal deformations of braided algebras.
Paper explores algebraic, topological, and combinatorial properties of singular virtual braids.
problem Understanding singular virtual braids and their properties.
method Algebraic relations, topological and combinatorial bijections, presentations.
result A bijection between singular abstract braids and singular virtual braids, leading to a presentation of the singular pure virtual braid monoid.
Study framizations of algebras using Schur--Weyl duality and tied braids.
problem Understanding framizations of algebras and their connections to quantum groups.
method Developing a general setting for framizations of algebras, including Yokonuma--Hecke and tied braids.
result Obtained Schur--Weyl duality for various algebras, including new framizations.
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending L∞-algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms. result Braided L∞-algebra is derived from the process. Researchers use Mellin-Barnes integrals to study trinomial equations and their braids.
problem Analyzing the roots of trinomial algebraic equations.
method Global analytic continuation and Mellin-Barnes integral representations.
result Precise description of the Galois group of trinomial equations.
This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this…
New algebraic structure helps distinguish braids.
problem Distinguishing braids using mathematical invariants.
method Defined pointed racks and used them to create braiding invariants.
result New invariants can distinguish braids not previously possible.
Braid groups help create complex surfaces in algebraic geometry.
problem Creating projective surfaces defined over complex numbers.
method Representation theory of higher genus braid groups.
result Interesting examples of projective surfaces produced.
New knot invariant from braided Hopf algebra.
problem Developing a new knot invariant.
method Non-commutative generalization of knot groups using braided Hopf algebra.
result New quantum character variety as an alternative to skein module.
New categories from TQFTs interpret skein relations.
problem Interpreting skein relations in TQFTs.
method Constructing half-braided algebras and their bimodules.
result Stated skein relations correspond to a TQFT.
Paper constructs representations for virtual braids and flat braids.
problem Calculating hyperbolic volumes of knot complements.
method Cluster algebra approach for virtual braid group.
result Forbidden relations do not hold in virtual braid group representation.
Study on virtual singular braid groups with algebraic properties and homomorphisms.
problem Algebraic properties and homomorphisms of virtual singular braid groups.
method Numerical invariants, homomorphisms, semi-direct product decompositions, presentations, and quotients.
result Determined all group homomorphisms from VSGn to Sn and obtained corresponding semi-direct product decompositions. The paper explores connections between braids, links, and cobordisms using algebraic methods.
problem Investigating functions on manifolds and their connections to braids, links, and cobordisms.
method Algebraic methods including group theory, sheaves, and formal groups.
result Constructs Lazard's one-dimensional universal commutative formal group and applies it to cobordism theory.
New algebra connects braid group actions to disc curves.
problem Understanding braid group actions on disc curves.
method Constructed a type B zigzag algebra and showed its categorical action on projective modules.
result Type B braid group action on homotopy category of projective modules.
In the 1920's Artin defined the braid group in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of th…
Explains fusion for Yang-Baxter equation and braid group.
problem Understanding algebras related to braid group, Yang-Baxter, and quantum groups.
method Introduces fusion procedure for Yang-Baxter equation.
result New examples of algebras: fused Hecke algebras.
In this paper we study the kernel of the homomorphism Bg,n→Bn of the braid group Bg,n in the handlebody Hg to the braid group Bn. We prove that this kernel is a semi-direct product of free groups. Also, we introduce an algebra Hg,n(q), which is some analog of the Hecke algebra $H_n(q…
The question of whether a representation of Artin's pure braid group is faithful is translated to certain properties of the Lie algebra arising from the descending central series of the pure braid group, and thus the Vassiliev invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main result is a Lie…
New braid group actions on n-adic integers linked to real algebraic links.
problem Understanding braid group actions on n-adic integers. method Constructing an infinite tower of covering spaces over configuration spaces and associating braids to infinite sequences of braids.
result An infinite family of braids close to real algebraic links.
The exterior algebra of a vector space admits a family of braided Hopf structures.
problem Identifying the exterior algebra with a Nichols algebra and studying its braided Hopf structures.
method Explicit computation of structure constants and construction of solutions to the Yang-Baxter equation.
result The exterior algebra of a vector space admits a one-parameter family of braided Hopf structures.
Maximal cubic quotient of braid algebra studied for n ≤ 5.
problem Understanding a maximal quotient of the braid group's algebra with cubic relations.
method Investigating quotients of the group algebra of the braid group with cubic relations.
result Proved isomorphism between the quotient and horizontal chord diagrams for n ≤ 5.
Study of bonded knots and braids with new algebraic models.
problem Classifying and understanding physical or chemical bonds in knots and braids.
method Developed new algebraic models (bonded knots, braids, braidoids) and invariants.
result New algebraic structures and invariants for bonded knots and braids.
Paper computes skein modules of 3-manifolds using braids.
problem Computing skein modules of 3-manifolds.
method Using braids and knot algebras.
result Braid approach to HOMFLYPT and Kauffman bracket skein modules of specific 3-manifolds.
New categorical actions link topological and algebraic structures.
problem Understanding relationships between topological and algebraic structures.
method Categorical actions of type B braid group on homotopy categories.
result Proves Rouquier's conjecture on faithfulness of Type B 2-braid group.
We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written x−1y where x and y are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type An and Bn and then show that in spherical …
Developed algebraic theory of bonded braids, proving Markov theorem.
problem Studied bonded knots and their algebraic properties.
method Introduced bonded braid monoid, proved Alexander and Markov theorems.
result Every bonded knot is the closure of a bonded braid, and equivalent knots have related braid representatives.
In this paper we define the p-adic framed braid group F∞,n, arising as the inverse limit of the modular framed braids and we give topological generators for F∞,n. We also give geometric interpretations for the p-adic framed braids. We then construct a p-adic Yokonuma-Hec…
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.
We determine the image of the braid groups inside the Temperley-Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under natural conditions on this parameter, the representations of the Hecke algebras …
New knot polynomials derived from Nichols algebras and braided Hopf algebras.
problem Developing new knot invariants from algebraic structures.
method Constructing knot invariants from solutions to the Yang--Baxter equation over generalized Yetter--Drinfel'd modules.
result Reproduces known knot polynomials and discovers new multivariable invariants.
New groups Γnk connect braids and 3-manifolds.
problem Understanding relationships between braids and 3-manifolds.
method Introducing and studying groups Γnk. result Groups Γn4 lead to new relationships between braids and manifolds. The paper constructs braiding structures for a specific subfactor.
problem The challenge is to understand the braiding structures of a Jones-Wassermann subfactor.
method The approach involves constructing braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra.
result The braiding structures induce a projective unitary representation of the balanced superelliptic mapping class group.
Proof confirms conjecture for certain braids and their closures.
problem Conjecture about real algebraic links and fibered links.
method Analyzes T-homogeneous and related braids, proving conjecture for their closures.
result Conjecture confirmed for closures of T-homogeneous braids.
In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this resu…
Proves knots in handlebodies can be represented as plats of braids.
problem Representing knots in handlebodies as plats of braids.
method Established Hilden braid group and mixed braid group, proving algebraic equivalence.
result Plat closure representation for knots in handlebodies.
We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion, orderability, linearity, and their relation both with mapping class groups and with the h…
Study of wild mapping class groups and their cabled braids.
problem Understanding the structure of wild mapping class groups and their cabled versions.
method Define and study generalizations of pure g-braid groups, establish product decompositions, and introduce fission trees. result Obtain cabled versions of braid groups, related to braid operads.
Research on knots, braids, and their invariants.
problem Understanding and computing knot invariants.
method Definition of knot equivalence, use of braid groups, Hecke algebras, and HOMFLY polynomial.
result General formula for HOMFLY polynomial of looped Coxeter braids.
We study a subset of square free positive braids and we give a few algebraic characterizations of them and one geometric characterization: the set of positive braids whose closures are unlinks. We describe canonical forms of these braids and of their conjugacy classes.
Formulates quantum jet bundles over noncommutative algebras with connections and braiding.
problem Defining jet bundles over noncommutative algebras with connections and braiding.
method Formalizes jet bundles over noncommutative algebras with flat connections and braiding tensor obeying Yang-Baxter equation.
result Examples include permutation groups, matrix algebras, and quantum spacetime models.
In this paper we describe braid equivalence for knots and links in a 3-manifold M obtained by rational surgery along a framed link in S3. We first prove a sharpened version of the Reidemeister theorem for links in M. We then give geometric formulations of the braid equivalence via mixed braids in S3 using the…
New findings on algebraic structure of hyperbolic graph braid groups.
problem Classifying and understanding the algebraic structure of hyperbolic graph braid groups.
method Analyzing specific graph types (sun and pulsar graphs) and proving theorems about their braid groups.
result 3-strand braid groups of sun graphs are free, while most pulsar graphs contain surface subgroups.
Study extends knot polynomials to links, identifying them with known invariants.
problem Extending knot polynomials to links.
method Applying Reshetikhin-Turaev functor to braided Hopf algebras with automorphisms.
result Identifies some knot polynomials with known link invariants.
We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev R-mat…
The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which can be represented as closed 3-braids. Partial results are obtained about spann…
We define an action of Artin's braid group on a finite dimensional algebra.