New representation for braid groups and surface braid groups, extending Lawrence-Krammer-Bigelow.
problem Constructing representations for braid groups and surface braid groups.
method Pro-nilpotent tower of representations, starting with the original LKB representation.
result 3-variable enrichment of the Lawrence-Krammer-Bigelow representation.
New braid representations using virtual knot theory.
problem No classical features in virtual knot theory.
method Construct new braid representations using virtual knot theory.
result New representations of classical braids.
Constructs six-dimensional braid group representations for knot detection.
problem Detecting braid vertibility of knots and links.
method Constructs six-dimensional block representations of B3 and uses them to detect braid vertibility. result Some representations can detect braid vertibility of known knots and links.
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.
Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We v…
Modified group captures braid dynamics, revealing Burau kernel.
problem Detecting the kernel of the Burau representation for braids.
method Defined a modified group Gn3 and its representation. result The modified group's representation detects the Burau kernel.
Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
Proves Burau representation is faithful for 4-strand braids.
problem Proving the Burau representation is faithful for 4-strand braids.
method Used ideas from Moody, Long, Long-Paton, and Bigelow.
result Burau representation is faithful for n = 4.
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
Extending braid group representations to singular braid monoids and groups.
problem Extending braid group representations to singular braid monoids and groups.
method Investigating the extension of representations from braid groups to singular braid monoids and groups, and computing defects.
result Constructing a linear representation of the singular braid group that is an extension of the Lawrence-Krammer-Bigelow representation and computing its defect.
The paper constructs representations of flat virtual braids by free group automorphisms.
problem Representing flat virtual braids by automorphisms of free groups.
method Construction of representations of flat virtual braid groups FVBn by automorphisms of free groups of rank 2n. result Established conditions of faithfulness and properties of the kernel for n≥3. Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
Researchers provide a simple topological method for Burau representations of loop braid groups.
problem Constructing Burau representations of loop braid groups.
method Simple topological construction of the Burau representations.
result One of the representations is more subtle and topologically natural but not easily combinatorially obvious.
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.
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…
Long and Moody gave a method of constructing representations of the braid group B_n. We discuss some ways to generalize their construction. One of these gives representations of subgroups of B_n, including the Gassner representation of the pure braid group as a special case. Another gives representations of the Hecke a…
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
Satellite links with many twists have simpler companions.
problem Relationship between satellite and companion links' complexity.
method Constructing satellite links with multiple full twists and analyzing their companion links.
result Satellite links with many twists have simpler companion links.
A new invariant for pure braids is defined and shown not to be trivial.
problem Defining a non-trivial invariant for pure braids.
method Using recoupling theory to define a representation of the pure braid group.
result The defined representation is not trivial.
Fast algorithm for braid group Hecke representation, applied to knot invariants.
problem Computing topological invariants of knots efficiently.
method Representation-theoretic approach to braid group, leveraging quantum topology.
result Fast algorithm for Hecke representation of braid group, finding non-trivial braids.
New braid group representations into mapping class groups are derived from disk coverings.
problem Tackling the representation of braid groups into mapping class groups.
method Explicit expressions of braid group representations into automorphism groups of free groups using covering groupoids.
result Each generator of braid group inside mapping class group is a product of Dehn twists.
We investigate a family of (reducible) representations of Artin's braid groups corresponding to a specific solution to the Yang-Baxter equation. The images of the braid groups under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the…
Paper uses Long-Moody construction for new braid group representations.
problem Constructing new representations of braid groups.
method Applies Fox derivation to matrix presentation of Long-Moody construction.
result Shows relation to twisted Alexander invariants.
New method computes knot invariants using free group automorphisms.
problem Computing knot invariants efficiently and accurately.
method Using representations of braid groups by automorphisms of a free group.
result Compared isotopic invariants to Alexander polynomials.
New polynomial invariants derived from birack and switch structures.
problem Polynomial invariants of braids.
method Switch structures, birack colorings, quiver-valued invariants.
result New polynomial invariants of braids.
Extends Lawrence's representations to integral Uqsl(2) Verma-modules and braid groups.
problem Integrating Lawrence's representations into Uqsl(2) Verma-modules and braid groups. method Defining homological operators and showing they provide a representation for Uqsl(2), establishing isomorphisms and preserving key properties. result Recovering an integral version of Kohno's theorem for Verma-modules and braid group representations.
By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup of the braid group…
Unified study of homological representations of mapping class groups.
problem Fundamental relationships between different homological representations.
method Analysis of twisted homology of configuration spaces.
result Unified account of non-degenerate pairings, embeddings, and isomorphisms.
We analyze two braid group representations and their reductions modulo p.
problem Analyzing the faithfulness of Gassner and Burau representations modulo p.
method Topological methods, including GL_n(Z[t_1^±1, ..., t_n^±1]) and GL_n(Z_p) reductions.
result The Gassner representation is faithful modulo p for all n and p > 1.
Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation being faithful.
Homotopy braid groups are shown to be torsion-free.
problem The existence of non-abelian torsion-free quotients of the braid group.
method Diagrammatic theory of welded braids and Artin representation.
result Homotopy braid groups are torsion-free.
Semisimplicity proven for conformal blocks representations.
problem Semisimplicity of conformal blocks representations.
method Theory of extensions in non-Abelian Hodge theory and Ocneanu rigidity.
result Semisimplicity of braid group and mapping class group representations.
We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one admitting an exchange move.
This is an expository article on diagrammatic representations of knots and links in various settings via braids.
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.
We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams a…
Generators found for a specific subgroup of braid groups.
problem Prove generators for the level m congruence subgroups of braid groups. method Proved using mth powers of half-twists and the braid Torelli group. result Solved Margalit's problem, generalizing previous work.
We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recent…
Link between braid groups and q-deformed rationals solves a classification problem.
problem Classifying faithful complex specializations of the Burau representation of braid group B3.
method Established a link between Burau representation and q-deformed rational numbers.
result Proved faithfulness of Burau representation outside a specific annulus.
We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Deh…
Introduction 1. The two-eigenvalue problem 2. Hecke algebra representations of braid groups 3. Duality of Jones-Wenzl representations 4. Closed images of Jones-Wenzl sectors 5. Distribution of evaluations of Jones polynomials 6. Fibonacci representations
Study on braid group quotients by congruence subgroups.
problem Understanding the image of congruence subgroups in GL(n,Z).
method Characterization through symplectic congruence subgroups.
result Open problem solved: image of congruence subgroups in GL(n,Z).
We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, …
A method to convert pretzel links into braids.
problem Representing pretzel links as braids.
method General procedure to convert any pretzel link into a braid.
result A specific braid word corresponding to the pretzel link.
The Burau representation of braid group B4 is shown to be faithful almost everywhere.
problem Demonstrate that the Burau representation of braid group B4 is faithful.
method Developed a combinatorial theory to explicitly determine Burau matrices, using Garside normal form and new product decompositions of positive braids. Used cancellation results to show faithfulness almost everywhere.
result The Burau representation of braid group B4 is faithful almost everywhere.
The Burau representation of 3-strand braid group modulo p is determined and shown to be faithful for small p.
problem Determining the faithfulness of the Burau representation of B3 modulo p. method Algorithm and proof for faithfulness, solving Salter's question for all p.
result The Burau representation of B3 modulo p is faithful for p≤13 and for all p. Paper connects algebraic and analytic methods for braid group representations.
problem Constructing representations of braid groups using algebraic and analytic approaches.
method Katz-Long-Moody construction and multiplicative middle convolution for KZ-type equations.
result Multiplicative middle convolution preserves unitarity and provides an algorithm to determine the signature of a Hermitian matrix.
The Lawrence representation Ln,m is a family of homological representation of the braid group Bn, which specializes to the reduced Burau and the Lawrence-Krammer representation when m is 1 and 2. In this article we show that the Lawrence representation is faithful for m≥2.