The paper connects knot homology to JM elements in affine braid groups.
problem Understanding knot homology through affine braid group elements.
method Constructing a homomorphism from affine braid group to knot homology.
result Derives a relation between knot homologies of specific braid group elements.
Categorical action of braid group on projective modules.
problem Action of extended braid group on projective modules.
method Constructed a trigraded quiver algebra to represent the group action on the homotopy category of projective modules.
result Intersection numbers from topological origin match trigraded dimensions of morphism spaces.
We show that the fundamental group of the space of ordered affine-equivalent configurations of at least five points in the real plane is isomorphic to the pure braid group modulo its centre. In the case of four points this fundamental group is free with eleven generators.
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.
The paper defines and analyzes configuration Lie groupoids and orbifold braid groups.
problem Understanding the structure and properties of orbifold braid groups.
method Definitions and proofs of fibration theorems, short exact sequences, and poly-virtually free structures.
result The pure orbifold braid groups have poly-virtually free structure, generalizing classical braid groups.
We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A_n, B_n=C_n and D_n and the affine diagrams tilde{A}_n, tilde{B}_n, tilde{C}_n and tilde{D}_n as subgroups of the braid gr…
We describe the fundamental groups of ordered and unordered k−point sets in the n-dimensional complex space Cn generating an affine subspace of fixed dimension.
Defines fundamental racks for braid spaces of complex reflection groups.
problem Understanding fundamental racks for braid spaces of complex reflection groups.
method Defines an augmented rack associated to the orbifold fundamental group.
result Yields representations of the orbifold fundamental group on the cohomology of the rack space.
We prove a long-standing conjecture about complex reflection arrangements.
problem The K(π,1) conjecture for affine Artin groups. method Recent advancements in dual Coxeter and Artin groups theory, new constructions, and poset shellability.
result The complexified complement of an affine reflection arrangement is a classifying space.
Affine Artin groups have a finite classifying space.
problem Proving the K(π,1) conjecture for affine Artin groups. method Dual Garside structures, Euclidean isometries, and shellability of noncrossing partitions.
result Affine Artin groups have a finite classifying space.
Maps between certain configuration spaces are rigid and affine equivalent.
problem Rigidity of maps between configuration spaces.
method Homomorphisms of braid groups and irreducibility conditions.
result Holomorphic maps between configuration spaces are affine equivalent.
Uniform proof of property R_infinity for specific Artin-Tits groups.
problem Establishing property R_infinity for various Artin-Tits groups.
method Uniform short proof for multiple groups using similar techniques.
result Property R_infinity confirmed for specified Artin-Tits groups.
Let G be one of the Artin groups of finite type Bn=Cn, and affine type A~n−1 and C~n−1. In this paper, we show that if α and β are elements of G such that αk=βk for some nonzero integer k, then α and β are conjugate in G. For the Artin …
Study on faithfulness of Burau representation for Artin-Tits groups.
problem Determining faithfulness of Burau representation for Artin-Tits groups.
method Algorithmic approach based on categorical methods.
result Burau representation is not faithful in specific cases.
New bounds on knot genus for positive braids.
problem Understanding the genus difference in positive braids.
method Analyzing the Seifert genus and 4-genus with respect to the number of strands.
result Characterizing knots with specific genus differences by forbidden minors.
Characterizes fractional Dehn twist coefficient and proves slice-Bennequin inequality.
problem Understanding the fractional Dehn twist coefficient and its relation to smooth slice genus.
method Characterization of FDTC and establishing the slice-Bennequin inequality.
result Affine linear lower bound for smooth slice genus in terms of FDTC.
In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingo…
The paper explores affine Hirsch foliations on 3-manifolds and classifies their existence.
problem Classifying the existence of affine Hirsch foliations on 3-manifolds.
method Analyzing and constructing 3-manifolds with specific foliations using exchangeable braided links.
result Every closed orientable 3-manifold admits 0, 1, or 2 affine Hirsch foliations, and every case is possible.
The paper reinterprets knot group invariants using affine transformations.
problem Alexander invariants of knots and their geometric interpretation.
method Representation varieties of knot groups into extrmAGL1(C). result Alexander polynomial as the singular locus of a coherent sheaf.
This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality…
Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain …
In this paper we construct effective invariants for braid monodromy of affine curves. We also prove that, for some curves, braid monodromy determines their topology. We apply this result to find a pair of curves with conjugate equations in a number field but which do not admit any orientation-preserving homeomorphism.
Our aim is to determine the lower central series (LCS) and derived series (DS) for the braid groups of the sphere and of the finitely-punctured sphere. We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string braid group B\_n(S^2) is constant from the commutator subgroup onwards, and that Γ\_2(B\_…
A mathematical isomorphism connects Floer homology to DAHA representations.
problem Connecting Floer homology to DAHA representations.
method Isomorphism between Floer homology and DAHA module structure.
result Establishes equivalence between DAHA polynomial representation and Floer homology module structure.
New construction shows DAAG/DAHA actions on congruence subgroups.
problem Understanding DAAG/DAHA actions on congruence subgroups.
method Coxeter-type presentations and adjoint DAAG/DAHA.
result DAAG/DAHA actions on congruence subgroups Γ1(r). Classic braids embed in virtual braids.
problem Embedding classic braids in virtual braids.
method Elementary proof using generalizations of braid groups.
result Classical braid group injects into virtual braid group.
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
Lectures on link homology and its algebraic/geometric models.
problem Defining and understanding link homology.
method Definition and properties of Khovanov-Rozansky triply graded homology, and three geometric models.
result Three geometric models for link homology: braid varieties, Hilbert schemes, and affine Springer fibers.
New connections between braid groups and free groups established.
problem Understanding relationships between different types of braid groups.
method Explicit homomorphisms between braid groups and free groups constructed.
result New invariants and complexities for classical braid groups defined.
The paper examines subgroup separability for surface and virtual braid groups.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
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.
The paper explores various braid-like groups and their properties.
problem Investigating the pure braid groups and their relatives.
method Examining resonance varieties, lower central series ranks, Chen ranks, residual and formality properties.
result Discussed natural homomorphisms and methods to distinguish braid-like groups.
Paper defines and equates loop braid groups, a mathematical concept.
problem Defining and equating loop braid groups.
method Introducing distinct approaches and proving their equivalence.
result Unified definitions of loop braid groups and their equivalence.
Study on groups of unrestricted virtual braids and their properties.
problem Understanding the structure and properties of virtual braids and links.
method Exploration of unrestricted virtual braids, fused links, flat virtual braids, and virtual Gauss braids.
result Definition and study of groups of flat virtual braids and virtual Gauss braids, including their linearity.
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
Introduces decorated braids for knot and braid theory.
problem Enhance braid theory with additional information at crossings.
method Introduces G-braids and Z2-braids, generalizing parity. result Dotted braid groups are a natural enhancement of classical braid groups.
Paper proves homotopy braid group properties over integers and three strands.
problem Understanding homotopy braid group properties.
method Proved linearity over integers and torsion freeness for three strands.
result Homotopy braid group on three strands is torsion free.
Polynomials with distinct critical values have braid monodromy groups equal to braid groups.
problem Understanding the structure of braid monodromy groups of polynomials.
method Analyzing the critical values of polynomials to determine their braid monodromy groups.
result The braid monodromy group of a polynomial equals the braid group if the polynomial has distinct critical values.
Study on commutator subgroups of singular and virtual braid groups.
problem Investigate commutator subgroups of singular and virtual braid groups.
method Analyzing the monoid and group structures of singular and virtual braid groups to prove their finitely generated and perfect properties.
result Prove that SGn′ is finitely generated if and only if n≥5, and GVBn′ is finitely generated if and only if n≥4. New virtual braid group structure and presentation found.
problem Understanding the structure of virtual pure braid groups.
method Cabling construction and HNN-extension approach.
result New presentation of VP4 and P4 groups. The paper studies orbifold braid groups and their properties.
problem Understanding orbifold braid groups and their subgroups.
method Detailed study of orbifold braid groups, proving injectivity and triviality of centers.
result Most orbifold braid groups have trivial centers.
We give formulae for the first homology of the n-braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first h…
We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective homomorphism. We prove that this map is a homomorphism for the braid groups of type…
New family of braided Thompson groups introduced using recursive braids.
problem Dehornoy-Brin braided Thompson group generalization.
method Using recursive braids and strand diagrams to define new groups.
result New groups BVn,r(H) are finitely generated if H is finitely generated. 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.
Classifies homomorphisms between specific braid groups.
problem Classifying homomorphisms between braid groups.
method Complete classification through recursive approach.
result Recursive classification of homomorphisms between braid groups.
Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to B3. result Quotients of congruence braid groups are almost crystallographic.